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Mechanisms and applications of the anti-inflammatory effects of photobiomodulation

  • Photobiomodulation (PBM) also known as low-level level laser therapy is the use of red and near-infrared light to stimulate healing, relieve pain, and reduce inflammation. The primary chromophores have been identified as cytochrome c oxidase in mitochondria, and calcium ion channels (possibly mediated by light absorption by opsins). Secondary effects of photon absorption include increases in ATP, a brief burst of reactive oxygen species, an increase in nitric oxide, and modulation of calcium levels. Tertiary effects include activation of a wide range of transcription factors leading to improved cell survival, increased proliferation and migration, and new protein synthesis. There is a pronounced biphasic dose response whereby low levels of light have stimulating effects, while high levels of light have inhibitory effects. It has been found that PBM can produce ROS in normal cells, but when used in oxidatively stressed cells or in animal models of disease, ROS levels are lowered. PBM is able to up-regulate anti-oxidant defenses and reduce oxidative stress. It was shown that PBM can activate NF-kB in normal quiescent cells, however in activated inflammatory cells, inflammatory markers were decreased. One of the most reproducible effects of PBM is an overall reduction in inflammation, which is particularly important for disorders of the joints, traumatic injuries, lung disorders, and in the brain. PBM has been shown to reduce markers of M1 phenotype in activated macrophages. Many reports have shown reductions in reactive nitrogen species and prostaglandins in various animal models. PBM can reduce inflammation in the brain, abdominal fat, wounds, lungs, spinal cord.

    Citation: Michael R Hamblin. Mechanisms and applications of the anti-inflammatory effects of photobiomodulation[J]. AIMS Biophysics, 2017, 4(3): 337-361. doi: 10.3934/biophy.2017.3.337

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  • Photobiomodulation (PBM) also known as low-level level laser therapy is the use of red and near-infrared light to stimulate healing, relieve pain, and reduce inflammation. The primary chromophores have been identified as cytochrome c oxidase in mitochondria, and calcium ion channels (possibly mediated by light absorption by opsins). Secondary effects of photon absorption include increases in ATP, a brief burst of reactive oxygen species, an increase in nitric oxide, and modulation of calcium levels. Tertiary effects include activation of a wide range of transcription factors leading to improved cell survival, increased proliferation and migration, and new protein synthesis. There is a pronounced biphasic dose response whereby low levels of light have stimulating effects, while high levels of light have inhibitory effects. It has been found that PBM can produce ROS in normal cells, but when used in oxidatively stressed cells or in animal models of disease, ROS levels are lowered. PBM is able to up-regulate anti-oxidant defenses and reduce oxidative stress. It was shown that PBM can activate NF-kB in normal quiescent cells, however in activated inflammatory cells, inflammatory markers were decreased. One of the most reproducible effects of PBM is an overall reduction in inflammation, which is particularly important for disorders of the joints, traumatic injuries, lung disorders, and in the brain. PBM has been shown to reduce markers of M1 phenotype in activated macrophages. Many reports have shown reductions in reactive nitrogen species and prostaglandins in various animal models. PBM can reduce inflammation in the brain, abdominal fat, wounds, lungs, spinal cord.


    The algebraic classification (up to isomorphism) of algebras of dimension n from a certain variety defined by a certain family of polynomial identities is a classic problem in the theory of non-associative algebras. There are many results related to the algebraic classification of small-dimensional algebras in many varieties of non-associative algebras [11,12,2,3,4,6,13,9,16]. So, algebraic classifications of 2-dimensional algebras [16,19], 3-dimensional evolution algebras [1], 3-dimensional anticommutative algebras [17], 4-dimensional division algebras [5,7], 4-dimensional nilpotent algebras [13] and 6-dimensional anticommutative nilpotent algebras [12] have been given. In the present paper, we give the algebraic classification of 5-dimensional nilpotent commutative algebras. The variety of commutative algebras is defined by the following identity: xy=yx. It contains commutative CD-algebras, Jordan algebras, mock-Lie algebras and commutative associative algebras as subvarieties. On the other hand, it is a principal part in the varieties of weakly associative algebras and flexible algebras.

    The algebraic study of central extensions of associative and non-associative algebras has been an important topic for years (see, for example, [10,20] and references therein). Our method for classifying nilpotent commutative algebras is based on the calculation of central extensions of nilpotent algebras of smaller dimensions from the same variety (first, this method has been developed by Skjelbred and Sund for Lie algebra case in [20]) and the classifications of all complex 5-dimensional nilpotent commutative (non-Jordan) CD-algebras [11]; nilpotent Jordan (non-associative) algebras [9]; and nilpotent associative commutative algebras [18].

    Throughout this paper, we use the notations and methods well written in [10], which we have adapted for the commutative case with some modifications. Further in this section we give some important definitions.

    Let (A,) be a complex commutative algebra and V be a complex vector space. The C-linear space Z2(A,V) is defined as the set of all bilinear maps θ:A×AV such that θ(x,y)=θ(y,x). These elements will be called cocycles. For a linear map f from A to V, if we define δf:A×AV by δf(x,y)=f(xy), then δfZ2(A,V). We define B2(A,V)={θ=δf :fHom(A,V)}. We define the second cohomology space H2(A,V) as the quotient space Z2(A,V)/B2(A,V).

    Let Aut(A) be the automorphism group of A and let ϕAut(A). For θZ2(A,V) define the action of the group Aut(A) on Z2(A,V) by

    ϕθ(x,y)=θ(ϕ(x),ϕ(y)).

    It is easy to verify that B2(A,V) is invariant under the action of Aut(A). So, we have an induced action of Aut(A) on H2(A,V).

    Let A be a commutative algebra of dimension m over C and V be a C-vector space of dimension k. For the bilinear map θ, define on the linear space Aθ=AV the bilinear product " [,]Aθ" by [x+x,y+y]Aθ=xy+θ(x,y) for all x,yA,x,yV. The algebra Aθ is called a k-dimensional central extension of A by V. One can easily check that Aθ is a commutative algebra if and only if θZ2(A,V).

    Call the set Ann(θ)={xA:θ(x,A)=0} the annihilator of θ. We recall that the annihilator of an algebra A is defined as the ideal Ann(A)={xA:xA=0}. Observe that Ann(Aθ)=(Ann(θ)Ann(A))V.

    The following result shows that every algebra with a non-zero annihilator is a central extension of a smaller-dimensional algebra.

    Lemma 1.1. Let A be an n-dimensional commutative algebra such that

    dim(Ann(A))=m0.

    Then there exists, up to isomorphism, a unique (nm)-dimensional commutative algebra A and a bilinear map θZ2(A,V) with Ann(A)Ann(θ)=0, where V is a vector space of dimension m, such that AAθ and A/Ann(A)A.

    Proof. Let A be a linear complement of Ann(A) in A. Define a linear map P:AA by P(x+v)=x for xA and vAnn(A), and define a multiplication on A by [x,y]A=P(xy) for x,yA. For x,yA, we have

    P(xy)=P((xP(x)+P(x))(yP(y)+P(y)))=P(P(x)P(y))=[P(x),P(y)]A.

    Since P is a homomorphism, P(A)=A and A is a commutative algebra and also A/Ann(A)A, which gives us the uniqueness. Now, define the map θ:A×AAnn(A) by θ(x,y)=xy[x,y]A. Thus, Aθ is A and therefore θZ2(A,V) and Ann(A)Ann(θ)=0.

    Definition 1.2. Let A be an algebra and I be a subspace of Ann(A). If A=A0I then I is called an annihilator component of A. A central extension of an algebra A without annihilator component is called a non-split central extension.

    Our task is to find all central extensions of an algebra A by a space V. In order to solve the isomorphism problem we need to study the action of Aut(A) on H2(A,V). To do that, let us fix a basis e1,,es of V, and θZ2(A,V). Then θ can be uniquely written as θ(x,y)=si=1θi(x,y)ei, where θiZ2(A,C). Moreover, Ann(θ)=Ann(θ1)Ann(θ2)Ann(θs). Furthermore, θB2(A,V) if and only if all θiB2(A,C). It is not difficult to prove (see [10,Lemma 13]) that given a commutative algebra Aθ, if we write as above θ(x,y)=si=1θi(x,y)eiZ2(A,V) and Ann(θ)Ann(A)=0, then Aθ has an annihilator component if and only if [θ1],[θ2],,[θs] are linearly dependent in H2(A,C).

    Let V be a finite-dimensional vector space over C. The Grassmannian Gk(V) is the set of all k-dimensional linear subspaces of V. Let Gs(H2(A,C)) be the Grassmannian of subspaces of dimension s in H2(A,C). There is a natural action of Aut(A) on Gs(H2(A,C)). Let ϕAut(A). For W=[θ1],[θ2],,[θs]Gs(H2(A,C)) define ϕW=[ϕθ1],[ϕθ2],,[ϕθs]. We denote the orbit of WGs(H2(A,C)) under the action of Aut(A) by Orb(W). Given

    W1=[θ1],[θ2],,[θs],W2=[ϑ1],[ϑ2],,[ϑs]Gs(H2(A,C)),

    we easily have that if W1=W2, then si=1Ann(θi)Ann(A)=si=1Ann(ϑi)Ann(A), and therefore we can introduce the set

    Ts(A)={W=[θ1],,[θs]Gs(H2(A,C)):si=1Ann(θi)Ann(A)=0},

    which is stable under the action of Aut(A).

    Now, let V be an s-dimensional linear space and let us denote by E(A,V) the set of all non-split s-dimensional central extensions of A by V. By above, we can write

    E(A,V)={Aθ:θ(x,y)=si=1θi(x,y)ei  and  [θ1],[θ2],,[θs]Ts(A)}.

    We also have the following result, which can be proved as in [10,Lemma 17].

    Lemma 1.3. Let Aθ,AϑE(A,V). Suppose that θ(x,y)=si=1θi(x,y)ei and ϑ(x,y)=si=1ϑi(x,y)ei. Then the commutative algebras Aθ and Aϑ are isomorphic if and only if

    Orb[θ1],[θ2],,[θs]=Orb[ϑ1],[ϑ2],,[ϑs].

    This shows that there exists a one-to-one correspondence between the set of Aut(A)-orbits on Ts(A) and the set of isomorphism classes of E(A,V). Consequently we have a procedure that allows us, given a commutative algebra A of dimension ns, to construct all non-split central extensions of A. This procedure is:

    1. For a given commutative algebra A of dimension ns, determine H2(A,C), Ann(A) and Aut(A).

    2. Determine the set of Aut(A)-orbits on Ts(A).

    3. For each orbit, construct the commutative algebra associated with a representative of it.

    The idea of the definition of a CD-algebra comes from the following property of Jordan and Lie algebras: the commutator of any pair of multiplication operators is a derivation. This gives three identities of degree four, which reduce to only one identity of degree four in the commutative or anticommutative case. Namely, a commutative algebra is a commutative CD-algebra (CCD-algebra) if it satisfies the following identity:

    ((xy)a)b+((xb)a)y+x((yb)a)=((xy)b)a+((xa)b)y+x((ya)b).

    The above described method gives all commutative (CCD- and non-CCD-) algebras. But we are interested in developing this method in such a way that it only gives non-CCD commutative algebras, because the classification of all CCD-algebras is done in [11]. Clearly, any central extension of a commutative non-CCD-algebra is a non-CCD-algebra. But a CCD-algebra may have extensions which are not CCD-algebras. More precisely, let D be a CCD-algebra and θZ2C(D,C). Then Dθ is a CCD-algebra if and only if

    θ(x,y)=θ(y,x),
    θ((xy)a,b)+θ((xb)a,y)+θ(x,(yb)a)=θ((xy)b,a)+θ((xa)b,y)+θ(x,(ya)b).

    for all x,y,a,bD. Define the subspace Z2D(D,C) of Z2C(D,C) by

    Z2D(D,C)={θZ2C(D,C):θ(x,y)=θ(y,x),θ((xy)a,b)+θ((xb)a,y)+θ(x,(yb)a)=θ((xy)b,a)+θ((xa)b,y)+θ(x,(ya)b) for all x,y,a,bD}.

    Observe that B2(D,C)Z2D(D,C). Let H2D(D,C)=Z2D(D,C)/B2(D,C). Then H2D(D,C) is a subspace of H2C(D,C). Define

    Rs(D)={WTs(D):WGs(H2D(D,C))},
    Us(D)={WTs(D):WGs(H2D(D,C))}.

    Then Ts(D)=Rs(D)Us(D). The sets Rs(D) and Us(D) are stable under the action of Aut(D). Thus, the commutative algebras corresponding to the representatives of Aut(D) -orbits on Rs(D) are CCD-algebras, while those corresponding to the representatives of Aut(D)-orbits on Us(D) are not CCD-algebras. Hence, we may construct all non-split commutative non-CCD-algebras A of dimension n with s-dimensional annihilator from a given commutative algebra A of dimension ns in the following way:

    1. If A is non-CCD, then apply the procedure.

    2. Otherwise, do the following:

    (a) Determine Us(A) and Aut(A).

    (b) Determine the set of Aut(A)-orbits on Us(A).

    (c) For each orbit, construct the commutative algebra corresponding to one of its representatives.

    Let us introduce the following notations. Let A be a nilpotent algebra with a basis e1,e2,,en. Then by Δij we will denote the bilinear form Δij:A×AC with Δij(el,em)=δilδjm, if ij and lm. The set {Δij:1ijn} is a basis for the linear space of bilinear forms on A, so every θZ2(A,V) can be uniquely written as θ=1ijncijΔij, where cijC. Let us fix complex number ηk (ηkk=1, ηlk1 for 0<l<k). For denote our algebras, we will use the following notations:

    NΞjjth5dimensional family ofcommutative nonCCDalgebras with parametrs Ξ.Nijjth idimensional nonCCDalgebra.Nijjth idimensional CCDalgebra.

    Remark 1. All families of algebras from our final list do not have intersections, but inside some families of algebras there are isomorphic algebras. All isomorphisms between algebras from a certain family of algebras constucted from the representative (Σ) are given in the list of distinct orbit representations. The notation (Ξ)O(Ξ1)=O(Ξ2) represents that the elements (Ξ1) and (Ξ2) have the same orbit.

    Thanks to [8] we have the complete classification of complex 4-dimensional nilpotent commutative algebras. It will be re-written by some different way for separating CCD- and non-CCD-algebras.

    N301,N401:e1e1=e2H2C=H2DN302,N402:e1e1=e2e1e2=e3H2CH2DN303,N403:e1e2=e3H2C=H2DN304,N404:e1e1=e2e2e2=e3H2CH2DN405:e1e1=e2e1e3=e4H2C=H2DN406:e1e1=e2e3e3=e4H2C=H2DN407:e1e1=e4e2e3=e4H2C=H2DN408:e1e1=e2e1e2=e3e2e2=e4H2CH2DN409:e1e1=e2e2e3=e4H2CH2DN410:e1e1=e2e1e2=e4e3e3=e4H2CH2DN411:e1e1=e2e1e3=e4e2e2=e4H2CH2DN412:e1e1=e2e2e2=e4e3e3=e4H2CH2DN413(λ):e1e1=e2e1e2=e3e1e3=e4e2e2=λe4H2CH2DN414:e1e2=e3e1e3=e4H2CH2DN415:e1e2=e3e1e3=e4e2e2=e4H2CH2DN416:e1e2=e3e1e3=e4e2e3=e4H2CH2DN417:e1e2=e3e3e3=e4H2CH2DN418:e1e1=e4e1e2=e3e3e3=e4H2CH2DN419:e1e1=e4e1e2=e3e2e2=e4e3e3=e4H2CH2DN401:e1e1=e2e1e2=e3e2e3=e4N402:e1e1=e2e1e2=e3e1e3=e4e2e3=e4N403:e1e1=e2e1e2=e3e3e3=e4N404:e1e1=e2e1e2=e3e2e2=e4e3e3=e4N405:e1e1=e2e1e3=e4e2e2=e3N406:e1e1=e2e1e2=e4e1e3=e4e2e2=e3N407:e1e1=e2e2e2=e3e2e3=e4N408:e1e1=e2e1e3=e4e2e2=e3e2e3=e4N409:e1e1=e2e2e2=e3e3e3=e4N410:e1e1=e2e2e2=e3e1e2=e4e3e3=e4N411(λ):e1e1=e2e1e2=λe4e2e2=e3e2e3=e4e3e3=e4

    Here we will collect all information about N302:

    CohomologyAutomorphismsN302e1e1=e2e1e2=e3H2D(N302)=[Δ13],[Δ22],H2C(N302)=H2D(N302)[Δ23],[Δ33]ϕ=(x00yx20z2xyx3)

    Let us use the following notations:

    1=[Δ13],2=[Δ22],3=[Δ23],4=[Δ33].

    Take θ=4i=1αiiH2C(N302). Since

    ϕT(00α10α2α3α1α3α4)ϕ=(ααα1αα2α3α1α3α4),

    we have

    α1=(α1x+α3y+α4z)x3,α2=(α2x2+4α3xy+4α4y2)x2,α3=(α3x+2α4y)x4,α4=α4x6.

    We are interested only in (α3,α4)(0,0) and consider the vector space generated by the following two cocycles:

    θ1=α11+α22+α33+α44  and  θ2=β11+β22+β33.

    Thus, we have

    α1=(α1x+α3y+α4z)x3,β1=(β1x+β3y)x3,α2=(α2x2+4α3xy+4α4y2)x2,β2=(β2x+4β3y)x3,α3=(α3x+2α4y)x4,β3=β3x5.α4=α4x6.

    Consider the following cases.

    1. α40, then:

    (a) β3=0,β20,β1=0, then by choosing x=2α24, y=α3α4, z=α232α1α4, we have the representatives 4,2;

    (b)  β3=0,β20,β10, then by choosing

    x=2α24β2, y=α3α4β2, z=α23(2β1+β2)+2α4(α2β1α1β2),

    we have the representatives 4,1+α2α0;

    (c) β3=0,β2=0,β10, then by choosing y=xα32α4, we have two representatives 4,1 and 2+4,1, depending on α23=α2α4 or not. The first representative will be joint with the family from the case (1b);

    (d) β30,4α2β23=4β2α3β3β22α4,β2=4β1, then by choosing

    x=4β3α4,y=β2α4,z=β2α34α1β3,

    we have the representative 4,3;

    (e) β30,4α2β23=4β2α3β3β22α4,β24β1, then by choosing

    x=4β1β24β3,y=β224β1β216β23,z=(4β1β2)(8β1α3β34β1β2α48α1β33+β22α4)32β33α4,

    we have the representative 4,1+3;

    (f) β30,4α2β234β2α3β3β22α4, then by choosing

    x=4α2β234β2α3β3+β22α44β23α4, y=β2α4β224α3β2β3+4α2β238β23α4, z=(8β1α3β34β1β2α48α1β33+β22α4)4α2β234β2α3β3+β22α416β33α4α4,

    we have the family of representatives 2+4,α1+3.

    2. α4=0,α30, then we may suppose that β3=0 and

    (a)  if β10,β2=4β1,α2=4α1, then by choosing x=α3,y=α1,z=0, we have the representative 3,1+42;

    (b) if β10,β2=4β1,α24α1, then by choosing x=α24α1α3,y=4α21α1α2α23,z=0, we have the representative 24(2+3),1+42;

    (c) if β10,β24β1, then by choosing x=α3(β24β1),y=β1α2α1β2,z=0, we have the family of representatives 3,1+α2α4, which will be jointed with the case (2a);

    (d) if β1=0, then we have the representative 33,2.

    Summarizing, we have the following distinct orbits:

    1,2+4,1+42,24(2+3),1+λ2,3,1+λ2,4,α1+3,2+4,1+3,4,2,33,2,4,3,4.

    Note that the algebras constructed from the orbits 1+42,24(2+3), 1+λ2,3, 1+α2,4, 2,33 and 2,4 are parts of some families of algebras which found below. Hence, we have the following new algebras:

    N12:e1e1=e2e1e2=e3e1e3=e4e2e2=e5e3e3=e5N4168:e1e1=e2e1e2=e3e1e3=e4e2e2=4e424e5e2e3=24e5Nλ,0170:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e2e3=e5Nλ,0184:e1e1=e2e1e2=e3e1e3=e4e2e2=λe4e3e3=e5Nα13:e1e1=e2e1e2=e3e1e3=αe4e2e2=e5e2e3=e4e3e3=e5N14:e1e1=e2e1e2=e3e1e3=e4e2e3=e4e3e3=e5N176:e1e1=e2e1e2=e3e2e2=e4e2e3=3e5N080:e1e1=e2e1e2=e3e2e2=e4e3e3=e5N15:e1e1=e2e1e2=e3e2e3=e4e3e3=e5

    Here we will collect all information about N304:

    N304e1e1=e2e2e2=e3H2D(N304)=[Δ12],H2C(N304)=H2D(N304)[Δ13],[Δ23],[Δ33]ϕ=(x000x20z0x4)

    Let us use the following notations:

    1=[Δ12],2=[Δ13],3=[Δ23],4=[Δ33].

    Take θ=4i=1αiiH2C(N304). Since

    ϕT(0α1α2α10α3α2α3α4)ϕ=(αα1α2α1αα3α2α3α4),

    we have

    α1=(α1x+α3z)x2,α2=(α2x+α4z)x4,α3=α3x5,α4=α4x8.

    Consider the following cases:

    1. α40, then consider the vector space generated by the following two cocycles:

    θ1=α11+α22+α33+α44  and  θ2=β11+β22+β33.

    Thus, we have

    α1=(α1x+α3z)x2,β1=(β1x+β3z)x2,α2=(α2x+α4z)x4,β2=β2x5,α3=α3x6,β3=β3x6.α4=α4x8.

    Then we consider the following subcases:

    (a) β3=0,α3=0, then we have:

    (i) if β1=0,α1=0, then we have the representative 4,2;

    (ii) if β1=0,α10, then by choosing x=5α1α41, we have the representative 1+4,2;

    (iii) if β10,β2=0, then by choosing x=1 and z=α2α14, we have the representative 4,1;

    (iv)  if β10,β20, then by choosing x=β1β21 and z=α1β2β1α2α4β1β2, we have the representative 4,1+2.

    (b) β3=0,α30, then we have:

    (i) if β2=0, then by choosing x=α3α41 and z=α2α3α34, we have the representative 3+4,1;

    (ii) if β20,β1=0, then by choosing

    x=α3α41 and z=α1α31α14,

    we have the representative 3+4,2;

    (iii) if β20,β10,β2α3=β1α4, then by choosing x=α3α41 and z=0, we have the family of representatives α1+3+4,1+2;

    (iv) if β20,β10,β2α3β1α4, then by choosing

    x=β1β2 and z=(α1β2β1α2)β1(β1α4β2α3)β2,

    we have the family of representatives α3+4,1+2α0,1, which will be jointed with the case (1(a)iv).

    (c) β30,α3=0, then we have:

    (i) if β2=0,α10, then by choosing x=5α1α14 and z=α25α1α64, we have the family of representatives 1+4,α1+3;

    (ii) if β2=0,α1=0, then by choosing z=α2xα4, we have two representatives 4,3 or 4,1+3 depending on whether β1α4=α2β3 or not;

    (iii) if β20, then by choosing x=β2β3 and z=α2β2β3α4, we have the family of representatives α1+4,β1+2+3.

    2. α4=0,α30, then we may suppose that β3=0. Thus, we have

    α1=(α1x+α3z)x2,β1=β1x3,α2=α2x5,β2=β2x5,α3=α3x6,

    and consider the following subcases:

    (a) β2=0, then we have two representatives 3,1 or 2+3,1, depending on whether α2=0 or not;

    (b) β20,α2=0, then by choosing z=α1xα3, we have two representatives 3,2 or 3,1+2, depending on whether β1=0 or not.

    3. α4=0,α3=0,β3=0,β2=0,α20, then we have the representative 2,1.

    Summarizing, we have the following distinct orbits:

    1,2, 1,2+3, 1,3, 1,3+4, 1,4, 1+2,α1+3+4O(α)=O(α), 1+2,3, 1+2,α3+4α1, β1+2+3,α1+4, α1+3,1+4O(α)=O(η3α)=O(η23α), 1+3,4, 1+4,2, 2,3, 2,3+4, 2,4, 3,4.

    Note that, the orbit 1,2 after a change of the basis of the constructed algebra gives a part of the family Nα79, which will be found below. Hence, we have the following new algebras:

    N076:e1e1=e2e1e2=e3e1e4=e5e2e2=e4N16:e1e1=e2e1e2=e4e1e3=e5e2e2=e3e2e3=e5N17:e1e1=e2e1e2=e4e2e2=e3e2e3=e5N18:e1e1=e2e1e2=e4e2e2=e3e2e3=e5e3e3=e5N19:e1e1=e2e1e2=e4e2e2=e3e3e3=e5Nα20:e1e1=e2e1e2=e4+αe5e1e3=e4e2e2=e3e2e3=e5e3e3=e5N21:e1e1=e2e1e2=e4e1e3=e4e2e2=e3e2e3=e5Nα122:e1e1=e2e1e2=e4e1e3=e4e2e2=e3e2e3=αe5e3e3=e5Nα,β23:e1e1=e2e1e2=βe4+αe5e1e3=e4e2e2=e3e2e3=e4e3e3=e5Nα24:e1e1=e2e1e2=αe4+e5e2e2=e3e2e3=e4e3e3=e5N25:e1e1=e2e1e3=e4e2e2=e3e2e3=e4e3e3=e5N26:e1e1=e2e1e2=e4e1e3=e5e2e2=e3e3e3=e4N27:e1e1=e2e1e3=e4e2e2=e3e2e3=e5N28:e1e1=e2e1e3=e4e2e2=e3e2e3=e5e3e3=e5N29:e1e1=e2e1e3=e4e2e2=e3e3e3=e5N30:e1e1=e2e2e2=e3e2e3=e4e3e3=e5

    Here we will collect all information about N402:

    N402e1e1=e2e1e2=e3H2D(N402)=[Δ13],[Δ22],[Δ14],[Δ24],[Δ44]H2C(N402)=H2D(N402)[Δ23],[Δ33],[Δ34]ϕ=(x000qx200w2xqx3re00t)

    Let us use the following notations:

    1=[Δ13],2=[Δ14],3=[Δ22],4=[Δ23],5=[Δ24],6=[Δ33],7=[Δ34],8=[Δ44].

    Take θ=8i=1αiiH2C(N402). Since

    ϕT(00α1α20α3α4α5α1α4α6α7α2α5α7α8)ϕ=(ααα1α2αα3α4α5α1α4α6α7α2α5α7α8),

    we have

    α1=(α1x+α4q+α6w+α7e)x3,α2=(α1x+α4q+α6w+α7e)r+(α2x+α5q+α7w+α8e)t,α3=(α3x2+4α4xq+4α6q2)x2,α4=(α4x+2α6q)x4,α5=(α4r+α5t)x2+2(α6r+α7t)xq,α6=α6x6,α7=(α6r+α7t)x3,α8=α6r2+2α7rt+α8t2.

    We interested in (α2,α5,α7,α8)(0,0,0,0) and (α4,α6,α7)(0,0,0). Let us consider the following cases:

    1. α6=0,α7=0, then α40 and we have the following subcases:

    (a) α8=0,α2α4α1α5=0, then we have a split extension;

    (b) α8=0,α2α4α1α50,α3=4α1, then by choosing

    x=4α2α4α1α5,t=α24,r=α4α5,q=α14α2α4α1α5α4,

    we have the representative 2+4;

    (c) α8=0,α2α4α1α50,α34α1, then by choosing

    x=α34α1α4,t=(α34α1)4α24(α2α4α1α5),r=α5(α34α1)4α34(α1α5α2α4),q=4α21α1α3α24,

    we have the representative 2+3+4;

    (d) α80,α3=4α1, then by choosing

    x=α4α8,t=α34α28,q=α1α8,r=α24α5α28,e=α1α5α2α4,

    we have the representative 4+8;

    (e) α80,α34α1, then by choosing

    x=α34α1α4,t=(α34α1)5α24α8,q=4α21α1α3α24,r=α5(α34α1)5α34α8,e=(4α1α3)(α2α4α1α5)α24α8,

    we have the representative 3+4+8.

    2. α6=0,α70, then we have the following subcases:

    (a) α4=0,α3=0, then by choosing

    x=2α27,q=α5α7,e=2α1α7,w=α25+2α1α82α2α7,t=2α7,r=α8,

    we have the representative 7;

    (b) α4=0,α30, then by choosing

    x=1, q=α52α7, e=α1α7, w=α25+2α1α82α2α72α27, t=α3α7, r=α3α82α27,

    we have the representative 3+7;

    (c) α40,α3α272α4α5α7+α24α8=0, then by choosing

    x=α7,t=α4,e=α34α14α7,r=α4α82α7,q=α3α74α4,w=4α1α4α84α2α4α7+α3(α5α7α4α8)4α4α37,

    we have the representative 4+7;

    (d) α40,α3α272α4α5α7+α24α80, then by choosing

    x=α32α4+α5α7α4α82α27,q=α3(α3α272α4α5α7+α24α8)8α24α27,w=(α3α272α4α5α7+α24α8)(4α2α4α74α1α4α8+α3(α5α7+α4α8))8α24α47,e=(4α1α3)(α3α272α4α5α7+α24α8)8α4α37,t=(α3α272α4α5α7+α24α8)24α4α57,r=α8(α3α272α4α5α7+α24α8)28α4α67,

    we have the representative 4+5+7.

    3. α60, then we have the following subcases:

    (a) α6α8α27=0,α5α6α4α7=0,α2α6α1α7=0, then we have a split extension;

    (b) α6α8α27=0,α5α6α4α7=0,α2α6α1α70,α3α6α24=0, then by choosing

    x=1,t=α26α2α6α1α7,q=α42α6,r=α6α7α1α7α2α6,e=0,w=α242α1α6α6,

    we have the representative 2+6;

    (c) α6α8α27=0,α5α6α4α7=0,α2α6α1α70,α3α6α240, then by choosing

    x=α3α6α24α26,t=(α3α6α24)5α36(α2α6α1α7),q=α4α3α6α242α26,r=(α3α6α24)5α7α46(α2α6α1α7),e=0,

    and w=(α242α1α6)α3α6α242α36, we have the representative 2+3+6;

    (d) α6α8α27=0,α5α6α4α70,2α6(α2α6α1α7)=α4(α5α6α4α7), then by choosing

    t=α26α5α6α4α7x4, q=α42α6x, r=α6α7α4α7α5α6x4, e=0, w=α242α1α6α6x,

    we have the representatives 5+6 and 3+5+6 depending on whether α3α6=α24 or not;

    (e) α6α8α27=0,α5α6α4α70,2α6(α2α6α1α7)α4(α5α6α4α7), then by choosing x=2α6(α2α6α1α7)α4(α5α6α4α7)2α26(α5α6α4α7), t=α26α5α6α4α7x4, q=α42α6x, r=α6α7α4α7α5α6x4, e=0, w=α242α1α6α6x, we have the representative 2+α3+5+6;

    (f) α6α8α270,α5α6α4α7=0, then by choosing

    t=α6x3α6α8α27,q=α4x2α6,r=α7x3α6α8α27,e=(α1α7α2α6)xα6α8α27,w=(α242α26+α1α8α2α7α27α6α8)x,

    we have the representatives 6+8 and 3+6+8 depending on whether α3α6=α24 or not.

    (g) α6α8α270,α5α6α4α70, then by choosing

    x=α5α6α4α7α26(α6α8α27),t=(α5α6α4α7)3α26(α27α6α8)2,q=α4(α4α7α5α6)2α6α26(α6α8α27),r=α7(α4α7α5α6)3α36(α27α6α8)2,e=α6(α5α6α4α7)(α4α5α6α24α7+2α6(α2α6+α1α7))2α36(α6α8α27)3,w=α6(α5α6α4α7)(α24α8α4α5α7+2α6(α2α7α1α8))2α36(α6α8α27)3,

    we have the representative α3+5+6+8.

    Summarizing, we have the following distinct orbits

    2+3+4,2+α3+5+6,2+3+6,2+4,2+6,3+4+8,3+5+6,α3+5+6+8,3+6+8,3+7,4+5+7,4+7,4+8,5+6,6+8,7,

    which gives the following new algebras:

    N31:e1e1=e2e1e2=e3e1e4=e5e2e2=e5e2e3=e5Nα32:e1e1=e2e1e2=e3e1e4=e5e2e2=αe5e2e4=e5e3e3=e5N33:e1e1=e2e1e2=e3e1e4=e5e2e2=e5e3e3=e5N34:e1e1=e2e1e2=e3e1e4=e5e2e3=e5N35:e1e1=e2e1e2=e3e1e4=e5e3e3=e5N36:e1e1=e2e1e2=e3e2e2=e5e2e3=e5e4e4=e5N37:e1e1=e2e1e2=e3e2e2=e5e2e4=e5e3e3=e5Nα38:e1e1=e2e1e2=e3e2e2=αe5e2e4=e5e3e3=e5e4e4=e5N39:e1e1=e2e1e2=e3e2e2=e5e3e3=e5e4e4=e5N40:e1e1=e2e1e2=e3e2e2=e5e3e4=e5N41:e1e1=e2e1e2=e3e2e3=e5e2e4=e5e3e4=e5N42:e1e1=e2e1e2=e3e2e3=e5e3e4=e5N43:e1e1=e2e1e2=e3e2e3=e5e4e4=e5N44:e1e1=e2e1e2=e3e2e4=e5e3e3=e5N45:e1e1=e2e1e2=e3e3e3=e5e4e4=e5N46:e1e1=e2e1e2=e3e3e4=e5

    Here we will collect all information about N404:

    N404e1e1=e2e2e2=e3H2D(N404)=[Δ12],[Δ14],[Δ24],[Δ44],H2C(N404)=H2D(N404)[Δ13],[Δ23],[Δ33],[Δ34]ϕ=(x0000x200y0x4rz00t)

    Let us use the following notations:

    1=[Δ12],2=[Δ13],3=[Δ14],4=[Δ23],5=[Δ24],6=[Δ33],7=[Δ34],8=[Δ44].

    Take θ=8i=1αiiH2C(N404). Since

    ϕT(0α1α2α3α10α4α5α2α4α6α7α3α5α7α8)ϕ=(αα1α2α3α1αα4α5α2α4α6α7α3α5α7α8),

    we have

    α1=(α1x+α4y+α5z)x2,α2=(α2x+α6y+α7z)x4,α3=(α2x+α6y+α7z)r+(α3x+α7y+α8z)t,α4=α4x6,α5=(α4r+α5t)x2,α6=α6x8,α7=(α6r+α7t)x4,α8=α6r2+2α7rt+α8t2.

    We interested in (α3,α5,α7,α8)(0,0,0,0) and (α2,α4,α6,α7)(0,0,0,0). Let us consider the following cases:

    1. α6=0,α7=0,α4=0, then α20 and we have the following cases:

    (a) if α8=0,α5=0, then by choosing t=1 and r=α3α2, we have a split extension;

    (b) if α8=0,α50, then by choosing

    x=α2α5, t=α42α25, z=α1α2, r=α32α3α25, y=0,

    we have the representative 2+5;

    (c) if α80,α5=0,α1=0, then by choosing

    x=α3, t=α2α28, z=α3, r=0, y=0,

    we have the representative 2+8;

    (d) if α80,α5=0,α10, then by choosing

    x=α1α21, t=4α51α32α18, z=α1α12α3α18, r=0, y=0,

    we have the representative 1+2+8;

    (e) if α80,α50, then by choosing

    x=α25α2α8,t=α55α22α38,z=α1α5α2α8,r=α45(α1α8α3α5)α32α38,y=0,

    we have the representative 2+5+8.

    2. α6=0,α7=0,α40, then by choosing r=α5α4t,y=α1x+α5zα4, we have α1=α5=0. Now we can suppose that α1=0,α5=0, and we have the following subcases:

    (a) if α8=0,α3=0, then we have a split extension;

    (b) if α8=0,α30,α2=0, then by choosing x=α3, y=0, z=0, r=0, t=α43, we have the representative 3+4;

    (c) if α8=0,α30,α20, then by choosing x=α2α4, y=0, z=0, r=0,t=α52α3α44, we have the representative 2+3+4;

    (d) if α80,α2=0, then by choosing x=1,y=0,z=α3α8,r=0,t=α4α8, we have the representative 4+8;

    (e) if α80,α20, then by choosing x=α3α4,y=0,z=α2α3α4α8,r=0,t=α32α54α8, we have the representative 2+4+8.

    3. α6=0,α70, then by choosing r=α8t2α7,y=(α2α8α3α7)xα27,z=α2xα7, we have α2=α3=α8=0. Now we can suppose that α2=0, α3=0, α8=0, and consider the following cases:

    (a) if α4=0,α5=0,α1=0, then we have the representative 7;

    (b) if α4=0,α5=0,α10, then by choosing x=1α7,t=α1,y=0,z=0,r=0, we have the representative 1+7;

    (c) if α4=0,α50,α1=0, then by choosing x=α5α7,t=1,y=0,z=0,r=0, we have the representative 5+7;

    (d) if α4=0,α50,α10, then by choosing x=α5α7,t=α21α5α7,y=0,z=0,r=0, we have the representative 1+5+7;

    (e) if α40,α5=0,α1=0, then by choosing x=α7,t=α4,y=0,z=0,r=0, we have the representative 4+7;

    (f) if α40,α5=0,α10, then by choosing x=3α1α4,t=3α1α24α37,y=0,z=0,r=0, we have the representative 1+4+7;

    (g) if α40,α50, then by choosing x=α5α7,t=α4α5α27,y=0,z=0,r=0, we have the representative α1+4+5+7.

    4. α60, then by choosing r=α7tα6,y=α2x+α7zα6, we have α2=α7=0. Now we can suppose that α2=0,α7=0, and we have:

    (a) if α8=0,α5=0, then α30 and we have the following subcases:

    (i) α4=0,α1=0, then by choosing x=α3,t=α63α6,y=0,z=0,r=0, we have the representative 3+6;

    (ii) α4=0,α10, then by choosing x=5α1α6,t=5α71α53α26,y=0,z=0,r=0, we have the representative 1+3+6;

    (iii) α40, then by choosing x=α4α6,t=5α74α23α56,y=0,z=0,r=0, we have the representative α1+3+4+6.

    (b) α8=0,α50, then we have the following subcases:

    (i) α4=0,α3=0, then by choosing x=6α5α6,t=1,z=α16α55α6,y=0,r=0, we have the representative 5+6;

    (ii) α4=0,α30, then by choosing x=α3α5,t=α63α75α6,z=α1α3α25,y=0,r=0, we have the representative 3+5+6;

    (iii) α40, then by choosing x=α4α6,t=α34α5α26,z=α1α15α4α16,y=0,r=0, we have the representative α3+4+5+6.

    (c) α80, then we have the following subcases:

    (i) α5=0,α4=0,α1=0, then by choosing x=1,t=α6α8,z=α3α8,y=0,r=0, we have the representative 6+8;

    (ii) α5=0,α4=0,α10, then by choosing

    x=5α1α16,t=10α81α36α58,z=α3α185α1α16,y=0,r=0,

    we have the representative 1+6+8;

    (iii) α5=0,α40, then by choosing

    x=α4α16, t=α24α36α18, z=α3α18α4α16, y=0, r=0,

    we have the representative α1+4+6+8;

    (iv) α50, then by choosing x=4α25α6α8, t=α25α6α38, z=α3α54α6α58, y=0, r=0, we have the representative α1+β4+5+6+8.

    Summarizing, we have the following distinct orbits:

    1+2+8,1+3+6,α1+3+4+6O(α)=O(α),α1+β4+5+6+8O(α,β)=O(α,β)=O(±iα,β),α1+4+6+8O(α)=O(α),1+4+7,α1+4+5+7O(α)=O(α),1+5+7,1+6+8,1+7,2+3+4,2+4+8,2+5,2+5+8,2+8,3+4,α3+4+5+6O(α)=O(α),3+5+6,3+6,4+7,4+8,5+6,5+7,6+8,7.

    Hence, we have the following new algebras:

    N47:e1e1=e2e1e2=e5e1e3=e5e2e2=e3e4e4=e5N48:e1e1=e2e1e2=e5e1e4=e5e2e2=e3e3e3=e5Nα49:e1e1=e2e1e2=αe5e1e4=e5e2e2=e3e2e3=e5e3e3=e5Nα,β50:e1e1=e2e1e2=αe5e2e2=e3e2e3=βe5e2e4=e5e3e3=e5e4e4=e5Nα51:e1e1=e2e1e2=αe5e2e2=e3e2e3=e5e3e3=e5e4e4=e5N52:e1e1=e2e1e2=e5e2e2=e3e2e3=e5e3e4=e5Nα53:e1e1=e2e1e2=αe5e2e2=e3e2e3=e5e2e4=e5e3e4=e5N54:e1e1=e2e1e2=e5e2e2=e3e2e4=e5e3e4=e5N55:e1e1=e2e1e2=e5e2e2=e3e3e3=e5e4e4=e5N56:e1e1=e2e1e2=e5e2e2=e3e3e4=e5N57:e1e1=e2e1e3=e5e1e4=e5e2e2=e3e2e3=e5N58:e1e1=e2e1e3=e5e2e2=e3e2e3=e5e4e4=e5N59:e1e1=e2e1e3=e5e2e2=e3e2e4=e5N60:e1e1=e2e1e3=e5e2e2=e3e2e4=e5e4e4=e5N61:e1e1=e2e1e3=e5e2e2=e3e4e4=e5N62:e1e1=e2e1e4=e5e2e2=e3e2e3=e5Nα63:e1e1=e2e1e4=αe5e2e2=e3e2e3=e5e2e4=e5e3e3=e5N64:e1e1=e2e1e4=e5e2e2=e3e2e4=e5e3e3=e5N65:e1e1=e2e1e4=e5e2e2=e3e3e3=e5N66:e1e1=e2e2e2=e3e2e3=e5e3e4=e5N67:e1e1=e2e2e2=e3e2e3=e5e4e4=e5N68:e1e1=e2e2e2=e3e2e4=e5e3e3=e5N69:e1e1=e2e2e2=e3e2e4=e5e3e4=e5N70:e1e1=e2e2e2=e3e3e3=e5e4e4=e5N71:e1e1=e2e2e2=e3e3e4=e5

    Here we will collect all information about N408:

    N408e1e1=e2e1e2=e3e2e2=e4H2D(N408)=[Δ13],[Δ14]+3[Δ23]H2C(N408)=H2D(N408)[Δ14],[Δ24],[Δ33],[Δ34],[Δ44]ϕ=(x000yx200z2xyx30ty2x2yx4)

    Let us use the following notations:

    1=[Δ13],2=[Δ14]+3[Δ23],3=[Δ14],4=[Δ24],5=[Δ33],6=[Δ34],7=[Δ44].

    Take θ=7i=1αiiH2C(N408). Since

    ϕT(00α1α2+α3003α2α4α13α2α5α6α2+α3α4α6α7)ϕ=(ααα1α2+α3αα3α2α4α13α2α5α6α2+α3α4α6α7),

    we have

    α1=(α1x+3α2y+α5z+α6t)x3+((α2+α3)x+α4y+α6z+α7t)x2y,α2=13(3α2x3+(α4+2α5)x2y+3α6xy2+α7y3)x2,α3=((α2+α3)x+α4y+α6z+α7t)x413(3α2x3+(α4+2α5)x2y+3α6xy2+α7y3)x2,α4=(α4x2+2α6xy+α7y2)x4,α5=(α5x2+2α6xy+α7y2)x4,α6=(α6x+α7y)x6,α7=α7x8.

    We are interested in (α3,α4,α5,α6,α7)(0,0,0,0,0), (α2+α3,α4,α6,α7)(0,0,0,0) and (α1,α2,α5,α6)(0,0,0,0). Let us consider the following cases:

    1. α7=0,α6=0,α5=0,α4=0, then α30, α2+α30 and (α1,α2)(0,0).

    (a)  if α2α34, then by choosing x=4α2+α3,y=α1, we have the representative α2+3α0,14,1;

    (b) if α2=α34, then we have the representatives

    142+3 and 1142+3

    depending on α1=0 or not.

    2. α7=0,α6=0,α5=0,α40, then by choosing y=3α2α4x, we have α2=0. This we can suppose α2=0, which implies α10 and choosing x=α1α14, we have the representative 1+α3+4.

    3. α7=0,α6=0,α50.

    (a)  if α4=0, then α2α3 and choosing

    x=α2+α3α5, y=3α2α3+3α232α25, z=(α2+α3)(2α1α5+12α2α3+3α23)4α35,

    we have the representative 2+5;

    (b) if α40,α4α5,2(α2α5α2α4+α3α5)+α3α4=0, then by choosing

    x=2(α4α5),y=3α3,z=0,t=0,

    we have the representative α4+5α0,1;

    (c)  if α40,α4α5,2(α2α5α2α4+α3α5)+α3α40, then by choosing

    x=2(α2α5α2α4+α3α5)+α3α42(α25α4α5),y=3α3(2(α2α5α2α4+α3α5)+α3α4)2α5(α5α4)2,z=(2α2(α4α5)α3(α4+2α5))(4α1(α4α5)224α2α3(α4α5)+3α23(α4+2α5))8(α4α5)3α25,t=0,

    we have the family of representatives 2+α4+5α0,1;

    (d) if α40,α4=α5, then by choosing y=α2xα5 and z=(α3α5α1α5+3α22)xα25, we have the representatives 4+5 and 3+4+5 depending on whether α3=0 or not. Note that 4+5=2+4+5 and it will be jointed with the family from the case (3c).

    4. if α7=0,α60, then by choosing x=1, y=(α4+2α5)236α2α6α42α56α6,

    z=y2α3α6+2y(α5α4)3α6 and t=x2α1+xy(4α2+α3)+xzα5+y(yα4+zα6)α6),

    we have α1=α2=α3=0. Now we can suppose that α1=0,α2=0,α3=0, and we have the following cases:

    (a) if α4=0,α5=0, then by choosing x=1,y=0,z=0,t=0, we have the representative 6;

    (b)  if α4=0,α50, then by choosing x=4α53α6,y=8α259α26,z=0,t=0, we have the representative 4+145+6;

    (c) if α40, then by choosing x=α4α6,y=0,z=0,t=0, we have the family of representatives 4+α5+6, which will be jointed with the representative from the case (4b).

    5. if α70, then by choosing x=1, y=α6α7, t=2α36+2(α4α5)α6α73α3α273α37 and z=0, we have α3=0,α6=0. Now we can suppose that α3=0,α6=0, and we have the following cases:

    (a) if α50, then by choosing x=α5α17,y=0,z=α21α15α17,t=0, we have the family of representatives α2+β4+5+7;

    (b) if α5=0,α2=0, then α10 and we have the family of representatives 1+α4+7;

    (c) if α5=0,α20, then by choosing x=3α2α17,y=0,z=0,t=0, we have the family of representatives α1+2+β4+7.

    Summarizing all cases we have the following distinct orbits

    1142+3,α1+2+β4+7O(α,β)=O(η3α,η23β)=O(η23α,η3β),1+α3+4O(α)=O(α),1+α4+7O(α)=O(α),α2+3α0,1,2+α4+5,α2+β4+5+7O(α,β)=O(α,β),3+4+5,α4+5α0,1,4+α5+6,6,

    which gives the following new algebras:

    N72:e1e1=e2e1e2=e3e1e3=e5e1e4=34e5e2e2=e4e2e3=34e5Nα,β73:e1e1=e2e1e2=e3e1e3=αe5e1e4=e5e2e2=e4e2e3=3e5e2e4=βe5e4e4=e5Nα74:e1e1=e2e1e2=e3e1e3=e5e1e4=αe5e2e2=e4e2e4=e5Nα75:e1e1=e2e1e2=e3e1e3=e5e2e2=e4e2e4=αe5e4e4=e5Nα0,176:e1e1=e2e1e2=e3e1e4=(1+α)e5e2e2=e4e2e3=3αe5Nα77:e1e1=e2e1e2=e3e1e4=e5e2e2=e4e2e3=3e5e2e4=αe5e3e3=e5Nα,β78:e1e1=e2e1e2=e3e1e4=αe5e2e2=e4e2e3=3αe5e2e4=βe5e3e3=e5e4e4=e5N79:e1e1=e2e1e2=e3e1e4=e5e2e2=e4e2e4=e5e3e3=e5Nα0,180:e1e1=e2e1e2=e3e2e2=e4e2e4=αe5e3e3=e5Nα81:e1e1=e2e1e2=e3e2e2=e4e2e4=e5e3e3=αe5e3e4=e5N82:e1e1=e2e1e2=e3e2e2=e4e3e4=e5

    Here we will collect all information about N409:

    N409e1e1=e2e2e3=e4H2D(N409)=[Δ12],[Δ13],[Δ22],[Δ33]H2C(N409)=H2D(N409)[Δ14],[Δ24],[Δ34],[Δ44]ϕ=(x0000x20000r0t0sx2r)

    Let us use the following notations:

    1=[Δ12],2=[Δ13],3=[Δ14],4=[Δ22],5=[Δ24],6=[Δ33],7=[Δ34],8=[Δ44].

    Take θ=8i=1αiiH2C(N409). Since

    ϕT(0α1α2α3α1α40α5α20α6α7α3α5α7α8)ϕ=(αα1α2α3α1α4αα5α1αα6α7α3α5α7α8),

    we have

    α1=(α1x+α5t)x2,α2=(α2x+α7t)r+(α3x+α8t)s,α3=(α3x+α8t)x2r,α4=α4x4,α5=α5x4r,α6=(α6r+α7s)r+(α7r+α8s)s,α7=(α7r+α8s)x2r,α8=α8r2x4.

    We are interested in (α3,α5,α7,α8)(0,0,0,0). Let us consider the following cases:

    1. α8=0,α7=0,α5=0, then α30 and we have

    (a) if α1=0,α4=0,α6=0, then by choosing x=1,r=α3,s=α2,t=0, we have the representative 3;

    (b) if α1=0,α4=0,α60, then by choosing x=α6,r=α3α26,s=α2α26,t=0, we have the representative 3+6;

    (c) if α1=0,α40,α6=0, then by choosing x=α23,r=α3α4,s=α2α4,t=0, we have the representative 3+4;

    (d) if α1=0,α40,α60, then by choosing x=α31α4α6, r=α32α34α6, s=α2α33α34α6, t=0, we have the representative 3+4+6;

    (e) if α10,α4=0,α6=0, then by choosing x=1, r=α1α31, s=α1α2α23, t=0, we have the representative 1+3;

    (f) if α10,α4=0,α60, then by choosing x=3α1α6α23, r=α1α13, s=α1α2α23, t=0, we have the representative 1+3+6;

    (g) if α10,α40, then by choosing

    x=α1α14, r=α1α13, s=α1α2α23, t=0,

    we have the family of representatives 1+3+4+α6.

    2. α8=0,α7=0,α50 and we have

    (a) if α3=0,α2=0,α4=0,α6=0, then by choosing r=1,x=α5,t=α1,s=0, we have the representative 5;

    (b) if α3=0,α2=0,α4=0,α60, then by choosing x=α5α6, r=α55α36,s=0,t=α1α6, we have the representative 5+6;

    (c) if α3=0,α2=0,α40,α6=0, then by choosing x=1,r=α4α51,t=α1α51,s=0, we have the representative 4+5;

    (d) if α3=0,α2=0,α40,α60, then by choosing

    x=4α4α6α25, r=α4α51, t=α14α4α6α65,s=0,

    we have the representative 4+5+6;

    (e) if α3=0,α20,α4=0,α6=0, then by choosing

    r=1,x=3α2α51,t=α13α2α45,s=0,

    we have the representative 2+5;

    (f) if α3=0,α20,α4=0,α60, then by choosing

    x=3α2α51, r=α163α42α15, t=α13α2α45,s=0,

    we have the representative 2+5+6;

    (g) if α3=0,α20,α40, then by choosing

    x=3α2α15,r=α4α51,t=α13α2α45,s=0,

    we have the family of representatives 2+4+5+α6;

    (h) if α30,α4=0,α6=0, then by choosing x=α3α51,r=α3,s=α2,t=α1α3α25, we have the representative 3+5;

    (i) if α30,α4=0,α60, then by choosing

    x=α3α51,r=α43α35α16,s=α2α33α35α16,t=α1α3α25,

    we have the representative 3+5+6;

    (j) if α30,α40, then by choosing x=α3α51,r=α4α51,s=α2α4α13α15,t=α1α3α25, we have the family of representatives 3+4+5+α6.

    3. α8=0,α70, then by choosing x=2α27, t=α3α62α2α7, s=α6, r=2α7, we have α2=α6=0. Now we can suppose that α2=0 and α6=0, then for s=0 and t=0, we have:

    (a) if α1=0,α3=0,α4=0,α5=0, then by choosing r=1,x=1, we have the representative 7;

    (b) if α1=0,α3=0,α4=0,α50, then by choosing x=α7,r=α5α7, we have the representative 5+7;

    (c) if α1=0,α3=0,α40,α5=0, then by choosing x=α7,r=α4, we have the representative 4+7;

    (d) if α1=0,α3=0,α40,α50, then by choosing x=α4α7α35,r=α4α51, we have the representative 4+5+7;

    (e) if α1=0,α30,α5=0, then by choosing r=α3,x=α5, we have the family of representatives 3+α4+7;

    (f) if α1=0,α30,α50, then by choosing x=α3α51,r=α23α15α17, we have the family of representatives 3+α4+5+7;

    (g) if α10,α3=0,α4=0,α5=0, then by choosing x=α1α7,r=α1, we have the representative 1+7;

    (h) if α10,α3=0,α4=0,α50, then by choosing x=3α1α7α25,r=3α21α15α17, we have the representative 1+5+7;

    (i) if α10,α3=0,α40, then by choosing x=α1α41,r=α1α14α17, we have the family of representatives 1+4+α5+7;

    (j) if α10,α30, then by choosing x=α1α7α23,r=α1α31, we have the family of representatives 1+3+α4+β5+7.

    4. α80, then by choosing x=α8,t=α3,s=α7,r=α8, we have α3=α7=0. Now we can suppose that α3=0 and α7=0, then for s=0 and t=0, we have

    (a) if α1=0,α2=0,α4=0,α6=0, then we have the representatives 8 and 5+8, depending on whether α5=0 or not;

    (b) if α1=0,α2=0,α4=0,α60, then we have the representatives 6+8 and 5+6+8, depending on whether α5=0 or not;

    (c) if α1=0,α2=0,α40,α6=0, then we have the representatives 4+8 and 4+5+8, depending on whether α5=0 or not;

    (d) if α1=0,α2=0,α40,α60, then by choosing x=4α6α81,r=α4α81, we have the representative 4+α5+6+8;

    (e) if α1=0,α20,α4=0,α5=0, then we have the representatives 2+8 and 2+6+8, depending on whether α6=0 or not;

    (f) if α1=0,α20,α4=0,α50, then by choosing x=3α2α51,r=α5α81, we have the representative 2+5+α6+8;

    (g) if α1=0,α20,α40, then by choosing x=6α22α14α18, r=α4α81, we have the representative 2+4+α5+β6+8;

    (h) if α10,α2=0,α4=0,α5=0, then we have the representatives 1+8 and 1+6+8 depending on whether α6=0 or not;

    (i) if α10,α2=0,α4=0,α50, then by choosing x=α1α8α25,r=α5α81, we have the representative 1+5+α6+8;

    (j) if α10,α2=0,α40, then by choosing x=α1α41,r=α4α81, we have the representative 1+4+α5+β6+8;

    (k) if α10,α20, then by choosing x=5α22α11α18,r=5α41α32α18, we have the representative 1+2+α4+β5+γ6+8.

    Summarizing, we have the following distinct orbits:

    1+2+α4+β5+γ6+8O(α,β,γ)=O(η5α,η35β,η5γ)=O(η25α,η5β,η25γ)=O(η35α,η45β,η35γ)=O(η45α,η25β,η45γ),1+3,1+3+α4+β5+7,1+3+4+α6,1+3+6,1+4+α5+β6+8O(α,β)=O(α,β),1+4+α5+7O(α,β)=O(α,β),1+5+α6+8,1+5+7,1+6+8,1+7,1+8,2+4+5+α6O(α)=O(η3α)=O(η23α),2+4+α5+β6+8,2+5,2+5+6,2+5+α6+8O(α,β)=O(α,β)=O(α,η23β)=O(α,η23β)=O(α,η3β)=O(α,η3β),2+6+8,2+8,3,3+4,3+4+5+α6,3+α4+5+7,3+4+6,3+5,3+5+6,3+6,4+5,4+5+6,4+α5+6+8O(α)=O(α),4+5+7,4+5+8,4+7,4+8,5,5+6,5+6+8,5+7,5+8,6+8,7,8,

    which gives the following new algebras:

    Nα,β,γ83:e1e1=e2e1e2=e5e1e3=e5e2e2=αe5e2e3=e4e2e4=βe5e3e3=γe5e4e4=e5N84:e1e1=e2e1e2=e5e1e4=e5e2e3=e4Nα,β85:e1e1=e2e1e2=e5e1e4=e5e2e2=αe5e2e3=e4e2e4=βe5e3e4=e5Nα86:e1e1=e2e1e2=e5e1e4=e5e2e2=e5e2e3=e4e3e3=αe5N87:e1e1=e2e1e2=e5e1e4=e5e2e3=e4e3e3=e5Nα,β88:e1e1=e2e1e2=e5e2e2=e5e2e3=e4e2e4=αe5e3e3=βe5e4e4=e5Nα89:e1e1=e2e1e2=e5e2e2=e5e2e3=e4e2e4=αe5e3e4=e5Nα90:e1e1=e2e1e2=e5e2e3=e4e2e4=e5e3e3=αe5e4e4=e5N91:e1e1=e2e1e2=e5e2e3=e4e2e4=e5e3e4=e5N92:e1e1=e2e1e2=e5e2e3=e4e3e3=e5e4e4=e5N93:e1e1=e2e1e2=e5e2e3=e4e3e4=e5N94:e1e1=e2e1e2=e5e2e3=e4e4e4=e5Nα95:e1e1=e2e1e3=e5e2e2=e5e2e3=e4e2e4=e5e3e3=αe5Nα,β96:e1e1=e2e1e3=e5e2e2=e5e2e3=e4e2e4=αe5e3e3=βe5e4e4=e5N97:e1e1=e2e1e3=e5e2e3=e4e2e4=e5N98:e1e1=e2e1e3=e5e2e3=e4e2e4=e5e3e3=e5Nα99:e1e1=e2e1e3=e5e2e3=e4e2e4=e5e3e3=αe5e4e4=e5N100:e1e1=e2e1e3=e5e2e3=e4e3e3=e5e4e4=e5N101:e1e1=e2e1e3=e5e2e3=e4e4e4=e5N102:e1e1=e2e1e4=e5e2e3=e4N103:e1e1=e2e1e4=e5e2e2=e5e2e3=e4Nα104:e1e1=e2e1e4=e5e2e2=e5e2e3=e4e2e4=e5e3e3=αe5Nα105:e1e1=e2e1e4=e5e2e2=αe5e2e3=e4e2e4=e5e3e4=e5N106:e1e1=e2e1e4=e5e2e2=e5e2e3=e4e3e3=e5N107:e1e1=e2e1e4=e5e2e3=e4e2e4=e5N108:e1e1=e2e1e4=e5e2e3=e4e2e4=e5e3e3=e5N109:e1e1=e2e1e4=e5e2e3=e4e3e3=e5N110:e1e1=e2e2e2=e5e2e3=e4e2e4=e5N111:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e3e3=e5Nα112:e1e1=e2e2e2=e5e2e3=e4e2e4=αe5e3e3=e5e4e4=e5N113:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e3e4=e5N114:e1e1=e2e2e2=e5e2e3=e4e2e4=e5e4e4=e5N115:e1e1=e2e2e2=e5e2e3=e4e3e4=e5N116:e1e1=e2e2e2=e5e2e3=e4e4e4=e5N117:e1e1=e2e2e3=e4e2e4=e5N118:e1e1=e2e2e3=e4e2e4=e5e3e3=e5N119:e1e1=e2e2e3=e4e2e4=e5e3e3=e5e4e4=e5N120:e1e1=e2e2e3=e4e2e4=e5e3e4=e5N121:e1e1=e2e2e3=e4e2e4=e5e4e4=e5N122:e1e1=e2e2e3=e4e3e3=e5e4e4=e5N123:e1e1=e2e2e3=e4e3e4=e5N124:e1e1=e2e2e3=e4e4e4=e5

    Here we will collect all information about N410:

    N410e1e1=e2e1e2=e4e3e3=e4H2D(N410)=[Δ13],[Δ14],[Δ22],[Δ23],[Δ33]H2C(N410)=H2D(N410)[Δ24],[Δ34],[Δ44]ϕ=(x000yx2zrx0z0r0tz2+2xysx3),r2=x3

    Let us use the following notations:

    1=[Δ13],2=[Δ14],3=[Δ22],4=[Δ23],5=[Δ24],6=[Δ33],7=[Δ34],8=[Δ44].

    Take θ=8i=1αiiH2C(N410). Since

    ϕT(00α1α20α3α4α5α1α4α6α7α2α5α7α8)ϕ=(ααα1α2αα3α4α5α1α4α6+αα7α2α5α7α8),

    we have

    α1=(α3y+α4z+α5t)zrx+(α1x+α4y+α6z+α7t)r+(α2x+α5y+α7z+α8t)s,α2=(α2x+α5y+α7z+α8t)x3,α3=α3x4+2α5x2(z2+2xy)+α8(z2+2xy)2,α4=(α3x2+α5(z2+2xy))zrx+(α4x2+α7(z2+2xy))r+(α5x2+α8(z2+2xy))s,α5=(α5x2+α8(z2+2xy))x3,α6=(α4rα3zrx+α5s)zrx+(α6rα4zrx+α7s)r+(α7rα5zrx+α8s)s(α3y+α4z+α5t)x2(α2x+α5y+α7z+α8t)(z2+2xy),α7=(α7rα5zrx+α8s)x3,α8=α8x6.

    We are interested in (α5,α7,α8)(0,0,0). Let us consider the following cases:

    1. α8=0,α5=0, then α70. Now by choosing

    y=α22+α2α3+α4α72α27x,z=α2α7x,s=3α22α3+α2(α23+6α4α7)+α7(α3α4+2α6α7)4α37x3,t=α27(α242α1α7)+α32α3+α22(α23+3α4α7)+2α2α7(α3α4+α6α7)2α47x,

    we have α1=0,α2=0,α4=0,α6=0. Then we have the representatives 7 or 3+7 depending on whether α3=0 or not.

    2. α8=0,α50, then by choosing

    y=α2α5+α27α25x,z=α7α5x,s=α3α7α4α5α25x3,t=α2α3α5+α25α6+3α4α5α72α3α27α35x,

    we have α2=α4=α6=0 and α7=0. Therefore, we can suppose that α2=0,α4=0,α6=0,α7=0, and have the following cases:

    (a) if α1=0,α3=0, then we have the representative 5;

    (b) if α1=0,α30, then by choosing x=α3α5,r2=x3, we have the representative 3+5;

    (c) if α10, then by choosing x=5α21α25,r2=x3, we have the family of representatives 1+α3+5.

    3. α80, then by choosing y=α5x2+α8z22α8x,s=x(α5zα7x)α8,t=α2x+α5y+α7zα8, we have α2=α5=α7=0. Therefore, we can suppose that α2=0,α5=0,α7=0, and have the following cases:

    (a) if α3=0,α4=0,α6=0, then we have the representative 8 and 1+8 depending on whether α1=0 or not.

    (b) if α3=0,α4=0,α60, then by choosing x=3α6α8,r2=x3,z=α13α26α8, we have the representative 6+8;

    (c) if α3=0,α40, then by choosing x=5α24α28,r2=x3,z=α635α34α28, we have the representative α1+4+8;

    (d) if α30, then by choosing x=α3α8,r2=x3,z=α4α3α8, we have the representative α1+3+β6+8.

    Summarizing, we have the following distinct orbits:

    1+α3+5O(α)=O(η45α)=O(η35α)=O(η25α)=O(η5α),α1+3+β6+8O(α,β)=O(α,β)=O(η3α,η23β)=O(η3α,η23β)=O(η23α,η3β)=O(η23α,η3β),α1+4+8O(α)=O(α)=O(η45α)=O(η45α)=O(η35α)=O(η35α)=O(η25α)=O(η25α)=O(η5α)=O(η5α),1+8,3+5,3+7,5,6+8,7,8,

    which gives the following new algebras:

    Nα125:e1e1=e2e1e2=e4e1e3=e5e2e2=αe5e2e4=e5e3e3=e4Nα,β126:e1e1=e2e1e2=e4e1e3=αe5e2e2=e5e3e3=e4+βe5e4e4=e5Nα127:e1e1=e2e1e2=e4e1e3=αe5e2e3=e5e3e3=e4e4e4=e5N128:e1e1=e2e1e2=e4e1e3=e5e3e3=e4e4e4=e5N129:e1e1=e2e1e2=e4e2e2=e5e2e4=e5e3e3=e4N130:e1e1=e2e1e2=e4e2e2=e5e3e3=e4e3e4=e5N131:e1e1=e2e1e2=e4e2e4=e5e3e3=e4N132:e1e1=e2e1e2=e4e3e3=e4+e5e4e4=e5N133:e1e1=e2e1e2=e4e3e3=e4e3e4=e5N134:e1e1=e2e1e2=e4e3e3=e4e4e4=e5

    Here we will collect all information about N411:

    N411e1e1=e2e1e3=e4e2e2=e4H2D(N411)=[Δ12],[Δ22],[Δ23],[Δ33]H2C(N411)=H2D(N411)[Δ14],[Δ24],[Δ34],[Δ44]ϕ=(x0000x200z0x30t2xzsx4)

    Let us use the following notations:

    1=[Δ12],2=[Δ14],3=[Δ22],4=[Δ23],5=[Δ24],6=[Δ33],7=[Δ34],8=[Δ44].

    Take θ=8i=1αiiH2C(N411). Since

    ϕT(0α10α2α1α3α4α50α4α6α7α2α5α7α8)ϕ=(αα1αα2α1α3+αα4α5αα4α6α7α2α5α7α8),

    we have

    α1=(α1x+α4z+α5t)x2+2(α2x+α7z+α8t)xz,α2=(α2x+α7z+α8t)x4,α3=(α3x2+4α5xz+4α8z2)x2(α6z+α7t)x3(α2x+α7z+α8t)s,α4=(α4x+2α7z)x4+(α5x+2α8z)xs,α5=(α5x+2α8z)x5,α6=α6x6+2α7x3s+α8s2,α7=(α7x3+α8s)x4,α8=α8x8.

    We are interested in (α2,α5,α7,α8)(0,0,0,0). Let us consider the following cases:

    1. α8=0,α7=0,α5=0, then α20 and we have

    (a)  if α4=0,α6=0, then by choosing x=2α2,z=α1,s=8α22α3,t=0, we have the representative 2;

    (b)  if α4=0,α60, then by choosing x=α2α6,z=α12α6,s=α1α2α6+2α22α32α36,t=0, we have the representative 2+6;

    (c) if α40,α4=2α2,α10, then by choosing

    x = \sqrt{\frac{\alpha_1}{\alpha_2}}, z = 0, s = \frac{\alpha_1\alpha_3\sqrt{\alpha_1}}{\alpha_2^2\sqrt{\alpha_2}}, t = 0,

    we have the family of representatives \langle \nabla_1+\nabla_2-2\nabla_4+\alpha\nabla_6 \rangle;

    {\rm{(d)}}\ if \alpha_4\neq0, \alpha_4 = -2\alpha_2, \alpha_1 = 0, \alpha_6\neq 0, then by choosing x = \alpha_2\alpha_6^{-1}, z = 0, s = \alpha_2^2\alpha_3\alpha_6^{-3}, t = 0, we have the representative \langle \nabla_2-2\nabla_4 +\nabla_6 \rangle;

    {\rm{(e)}}\ if \alpha_4\neq0, \alpha_4 = -2\alpha_2, \alpha_1 = 0,\alpha_6 = 0, then by choosing x = \alpha_2, z = 0, s = \alpha_2^2\alpha_3, t = 0, we have the representative \langle \nabla_2-2\nabla_4 \rangle;

    {\rm{(f)}}\ if \alpha_4\neq0, \alpha_4\neq-2\alpha_2, \alpha_6 = 0, then by choosing

    x = \alpha_4+2\alpha_2, z = -\alpha_1, s = \frac{\alpha_3(\alpha_4+2\alpha_2)^3}{\alpha_2}, t = 0,

    we have the the family of representatives \langle \nabla_2+\alpha\nabla_4 \rangle_{\alpha\neq0,-2}, which will be jointed with the cases (1a) and (1b);

    {\rm{(g)}}\ if \alpha_4\neq0, \alpha_4\neq-2\alpha_2, \alpha_6\neq0, then by choosing

    x = \frac{\alpha_2}{\alpha_6}, z = -\frac{\alpha_1\alpha_2}{\alpha_6(\alpha_4+2\alpha_2)}, s = \frac{\alpha_1\alpha^2_2\alpha_6+2\alpha_2^3\alpha_3+\alpha_2^2\alpha_3\alpha_4}{\alpha_6^3(\alpha_4+2\alpha_2)}, t = 0,

    we have the family of representatives \langle \nabla_2+\alpha\nabla_4+\nabla_6 \rangle_{\alpha\neq0,-2}, which will be jointed with the cases (1b) and (1d).

    2. \alpha_8 = 0, \alpha_7 = 0, \alpha_5\neq0 , then we have

    {\rm{(a)}}\ if \alpha_6 = 0, \alpha_2 = 0, then by choosing

    x = 4\alpha_5, z = -\alpha_3, s = -64\alpha_4\alpha_5^2, t = \frac{\alpha_3\alpha_4-4\alpha_1\alpha_5}{\alpha_5}

    we have the representative \langle \nabla_5 \rangle;

    {\rm{(b)}}\ if \alpha_6 = 0, \alpha_2\neq0, then by choosing

    \begin{array}{c} x = \frac{\alpha_2}{\alpha_5}, z = -\frac{\alpha_2^2\alpha_4+\alpha_2\alpha_3\alpha_5}{4\alpha_5^3}, s = -\frac{\alpha_2^3\alpha_4}{\alpha_5^4}, \\ t = \frac{(\alpha_2\alpha_4+2\alpha^2_2)(\alpha_2\alpha_4+\alpha_3\alpha_5)-4\alpha_1\alpha_2\alpha_5^2}{4\alpha_5^4},\end{array}

    we have the representative \langle \nabla_2+\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_6\neq0, \alpha_6 = 4\alpha_5, \alpha_3 = 0, \alpha_2 = 0, then by choosing x = \alpha_5, z = 0, s = -\alpha_4\alpha_5^2, t = -\alpha_1, we have the representative \langle \nabla_5+4\nabla_6 \rangle;

    {\rm{(d)}}\ if \alpha_6\neq0, \alpha_6 = 4\alpha_5, \alpha_2\alpha_4+\alpha_3\alpha_5 = 0, \alpha_2\neq0, then by choosing

    x = \frac{\alpha_2}{\alpha_5}, z = 0, s = -\frac{\alpha_2^3\alpha_4}{\alpha_5^4}, t = -\frac{\alpha_1\alpha_2}{\alpha_5^2},

    we have the representative \langle \nabla_2+\nabla_5+4\nabla_6 \rangle;

    {\rm{(e)}}\ if \alpha_6\neq0, \alpha_6 = 4\alpha_5, \alpha_2\alpha_4+\alpha_3\alpha_5\neq0, then by choosing

    x = \frac{\sqrt{\alpha_2\alpha_4+\alpha_3\alpha_5}}{\alpha_5}, z = 0, s = -\frac{\alpha_4\sqrt{(\alpha_2\alpha_4+\alpha_3\alpha_5)^3}}{\alpha^4_5}, t = -\frac{\alpha_1\sqrt{\alpha_2\alpha_4+\alpha_3\alpha_5}}{\alpha^2_5},

    we have the family of representatives \langle \alpha\nabla_2+\nabla_3+\nabla_5+4\nabla_6 \rangle;

    {\rm{(f)}}\ if \alpha_6\neq0, \alpha_6\neq4\alpha_5, \alpha_2 = 0, then by choosing

    x = \alpha_6-4\alpha_5, z = \alpha_3, s = \frac{\alpha_4(4\alpha_5-\alpha_6)^3}{\alpha_5}, t = \frac{4\alpha_1\alpha_5-\alpha_1\alpha_6-\alpha_3\alpha_4}{\alpha_5},

    we have the family of representatives \langle \nabla_5+\alpha\nabla_6 \rangle_{\alpha\neq0,4}, which will be jointed with the cases (2a) and (2c);

    {\rm{(g)}}\ if \alpha_6\neq0, \alpha_6\neq4\alpha_5, \alpha_2\neq0, then by choosing

    \begin{array}{c} x = \frac{\alpha_2}{\alpha_5}, z = \frac{\alpha_2 (\alpha_2 \alpha_4+\alpha_3 \alpha_5)}{\alpha_5^2\alpha_6-4\alpha_5^2}, s = -\frac{\alpha_4\alpha_2^3}{\alpha^4_5}, \\ t = \frac{\alpha_2 (2 \alpha_2^2 \alpha_4+\alpha_3 \alpha_4 \alpha_5+\alpha_2 (\alpha_4^2+2 \alpha_3 \alpha_5)-\alpha_1 \alpha_5 (4 \alpha_5-\alpha_6))}{\alpha_5^3 (4 \alpha_5-\alpha_6)}, \end{array}

    we have the family of representatives \langle \nabla_2+\nabla_5+\alpha\nabla_6 \rangle_{(\alpha\neq0,4)}, which will be jointed with the cases (2a) and (2c).

    3. \alpha_8 = 0, \alpha_7\neq0 , then by choosing z = -\frac{\alpha_2}{\alpha_7}x, s = -\frac{\alpha_6}{2\alpha_7}x^3, t = \frac{ \alpha_2 (\alpha_6-4 \alpha_5)+\alpha_3 \alpha_7}{\alpha^2_7}x, we have \alpha_2^* = \alpha_3^* = \alpha_6^* = 0 . Now we can suppose that \alpha_2 = 0, \alpha_3 = 0, \alpha_6 = 0 and have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_5 = 0, then we have the representative \langle \nabla_7 \rangle;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_5\neq0, then by choosing x = \frac{\alpha_5}{\alpha_7}, z = 0, s = 0, t = 0, we have the representative \langle \nabla_5+\nabla_7 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_4\neq0, then by choosing x = \sqrt{\frac{\alpha_4}{\alpha_7}}, z = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\nabla_7 \rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_1}{\alpha_7}}, z = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_4+\beta\nabla_5+\nabla_7 \rangle.

    4. \alpha_8\neq0 , then by choosing z = -\frac{\alpha_5}{2\alpha_8}x, s = -\frac{\alpha_7}{\alpha_8}x^3, t = \frac{\alpha_5\alpha_7-2\alpha_2\alpha_8}{2\alpha^2_8}x, we have \alpha_2^* = \alpha_5^* = \alpha_7^* = 0 . Now we can suppose that \alpha_2 = 0, \alpha_5 = 0, \alpha_7 = 0 and have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, \alpha_6 = 0, then we have the representative \langle \nabla_8 \rangle;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, \alpha_6\neq0, then by choosing x = \sqrt{\frac{\alpha_6}{\alpha_8}}, z = 0, s = 0, t = 0, we have the representative \langle \nabla_6+\nabla_8 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_4}{\alpha_8}}, z = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_4+\alpha\nabla_6+\nabla_8 \rangle;

    {\rm{(d)}}\ if \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_3}{\alpha_8}}, z = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle;

    {\rm{(e)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[5]{\frac{\alpha_1}{\alpha_8}}, z = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\nabla_8 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+ \nabla_2 - 2 \nabla_4 + \alpha\nabla_6\rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_1+ \alpha\nabla_3 +\beta \nabla_4+\gamma \nabla_6 + \nabla_8 \rangle^{ { \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\eta_5\alpha,\eta_5^2\beta,-\eta_5^3\gamma) = O(\eta_5^2\alpha,\eta_5^4\beta,-\eta_5\gamma) = \\ O(-\eta_5^3\alpha,-\eta_5\beta,\eta_5^4\gamma) = O(\eta_5^4\alpha,-\eta_5^3\beta,\eta_5^2\gamma)\end{array}}}, \\ \langle \nabla_1+\alpha\nabla_4+\beta\nabla_5+\nabla_7 \rangle ^{O(\alpha,\beta) = O(\alpha,-\beta) = O(-\alpha,-i\beta) = O(-\alpha,i\beta)}, \\ \langle \alpha\nabla_2+ \nabla_3+\nabla_5 + 4\nabla_6 \rangle^{O(\alpha) = O(-\alpha)}, \langle \nabla_2+ \alpha\nabla_4 \rangle, \langle \nabla_2+ \alpha\nabla_4 + \nabla_6 \rangle, \\ \langle \nabla_2+ \nabla_5 + \alpha\nabla_6 \rangle, \langle \nabla_3+ \alpha\nabla_4+\beta\nabla_6 + \nabla_8 \rangle, \langle \nabla_4+ \alpha\nabla_5+\nabla_7 \rangle, \\ \langle \nabla_4+ \alpha\nabla_6+\nabla_8 \rangle, \langle \nabla_5+ \alpha\nabla_6 \rangle, \langle \nabla_5+ \nabla_7 \rangle, \langle \nabla_6+ \nabla_8 \rangle, \langle \nabla_7 \rangle, \langle \nabla_8 \rangle,\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{135}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{136}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_4+\alpha e_5 \\ & & e_2e_3 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{137}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{138}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = 4e_5 \\ {\mathbf{N}}_{139}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{140}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{141}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\& & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{142}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4+e_5 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{143}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{144}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{145}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{146} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{147} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{148} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{149} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{12}^{4*}:

    \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{12} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_2 = e_4 \\ e_3e_3 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{12})& = &\\ {\langle [\Delta_{12}],[\Delta_{13}],[\Delta_{23}],[\Delta_{33}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{12})& = &\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{12})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle } \end{array} & \phi_{\pm} = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0& \pm x^2&0\\ t&0&s&x^4 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{8}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{12}) . Since

    \phi_{\pm}^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&\alpha_4&\alpha_5\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7\\ \alpha_3&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi_{\pm} = \begin{pmatrix} \alpha^*&\alpha_1^{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha_1^{*}&0&\alpha^*_4&\alpha^*_5\\ \alpha^{*}_2&\alpha^*_4&\alpha^*_6&\alpha^*_7\\ \alpha^*_3&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix},

    we have

    \begin{array}{ll} \alpha_1^* = (\alpha_1x+\alpha_5t)x^2, & \alpha_2^* = (\alpha_3x+\alpha_8t)s\pm(\alpha_2x+\alpha_7t)x^2, \\ \alpha_3^* = (\alpha_3x+\alpha_8t)x^4, & \alpha_4^* = (\alpha_5s\pm\alpha_4x^2)x^2, \\ \alpha_5^* = \alpha_5x^6, & \alpha_6^* = \alpha_6x^4\pm2\alpha_7sx^2+\alpha_8s^2, \\ \alpha_7^* = (\alpha_8s\pm\alpha_7x^2)x^4, & \alpha_8^* = \alpha_8x^8. \end{array}

    We will consider only the action of \phi_+ for find representatives and after that we will see that the set of our representatives gives distinct orbits under action of \phi_+ and \phi_-. We are interested in (\alpha_3,\alpha_5,\alpha_7,\alpha_8)\neq(0,0,0,0) . Let us consider the following cases:

    1.\ \alpha_8 = 0, \alpha_5 = 0, \alpha_7 = 0, then \alpha_3\neq0 and we have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_6 = 0, then by choosing x = \alpha_3, s = -\alpha_2\alpha_3, t = 0, we have the representative \langle \nabla_3 \rangle;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_6\neq0, then by choosing x = \frac{\alpha_6}{\alpha_3}, s = -\frac{\alpha_2\alpha_6^2}{\alpha_3^3}, t = 0, we have the representative \langle \nabla_3+\nabla_6 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_4\neq0, then by choosing x = \frac{\alpha_4}{\alpha_3}, s = -\frac{\alpha_2\alpha^2_4}{\alpha_3^3}, t = 0, we have the representative \langle \nabla_3+\nabla_4+\alpha\nabla_6 \rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, then by choosing x = \sqrt{\frac{\alpha_1}{\alpha_3}}, s = -\frac{\alpha_1\alpha_2}{\alpha_3^2}, t = 0, we have the representative \langle \nabla_1+\nabla_3+\alpha\nabla_4+\beta\nabla_6 \rangle.

    2. \alpha_8 = 0, \alpha_5 = 0, \alpha_7\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, then by choosing x = 1, s = -\frac{\alpha_6}{2\alpha_7}, t = \frac{\alpha_3\alpha_6-2\alpha_2\alpha_7}{2\alpha_7^2}, we have the representative \langle \nabla_7 \rangle;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt{\frac{\alpha_4}{\alpha_7}}, s = -\frac{\alpha_4\alpha_6}{2\alpha^2_7}, t = \frac{\sqrt{\alpha_4}(\alpha_3\alpha_6-2\alpha_2\alpha_4)}{2\alpha_7^2\sqrt{\alpha_7}}, we have the representative \langle \nabla_4+\nabla_7 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \frac{\alpha_3}{\alpha_7}, s = -\frac{\alpha^2_3\alpha_6}{2\alpha^3_7}, t = \frac{\alpha_3^2\alpha_6-2\alpha_2\alpha_3\alpha_7}{2\alpha_7^3}, we have the representative \langle \nabla_3+\alpha\nabla_4+\nabla_7 \rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_1}{\alpha_7}}, s = -\sqrt[3]{\frac{\alpha_1^2\alpha^3_6}{8\alpha^5_7}}, t = \frac{\sqrt[3]{\alpha_1}(\alpha_3\alpha_6-2\alpha_2\alpha_7)}{2\alpha_7^2\sqrt[3]{\alpha_7}}, we have the representative \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\nabla_7 \rangle.

    3. \alpha_8 = 0, \alpha_5\neq0, then by choosing t = -\frac{\alpha_1}{\alpha_5}x, s = -\frac{\alpha_4}{\alpha_5}x^2, we have \alpha_1^* = \alpha_4^* = 0. Now we can suppose that \alpha_1 = 0, \alpha_4 = 0 and have the following subcases:

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_3 = 0, \alpha_6 = 0, then we have the family of representatives \langle \nabla_5+\alpha\nabla_7 \rangle;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_3 = 0, \alpha_6\neq0, then by choosing x = \sqrt{\frac{\alpha_6}{\alpha_5}}, s = 0, t = 0, we have the family of representatives \langle \nabla_5+\nabla_6+\alpha\nabla_7 \rangle;

    {\rm{(c)}}\ if \alpha_2 = 0, \alpha_3\neq0, then by choosing x = \frac{\alpha_3}{\alpha_5}, s = 0, t = 0, we have the family of representatives \langle \nabla_3+\nabla_5+\alpha\nabla_6+\beta\nabla_7 \rangle;

    {\rm{(d)}}\ if \alpha_2\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_2}{\alpha_5}}, s = 0, t = 0, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+\nabla_5+\beta\nabla_6+\gamma\nabla_7 \rangle.

    4. \alpha_8\neq0, then by choosing t = -\frac{\alpha_3}{\alpha_8}x, s = -\frac{\alpha_7}{\alpha_8}x^2, we have \alpha_3^* = \alpha_7^* = 0. Now we can suppose that \alpha_3 = 0, \alpha_7 = 0 and have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_4 = 0, \alpha_5 = 0, \alpha_6 = 0, then we have the representative \langle \nabla_8 \rangle;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_4 = 0, \alpha_5 = 0, \alpha_6\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_6}{\alpha_8}}, s = 0, t = 0, we have the representative \langle \nabla_6+\nabla_8 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_4 = 0, \alpha_5\neq0, then by choosing x = \sqrt{\frac{\alpha_5}{\alpha_8}}, s = 0, t = 0, we have the family of representatives \langle \nabla_5+\alpha\nabla_6+\nabla_8 \rangle;

    {\rm{(d)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_4\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_4}{\alpha_8}}, s = 0, t = 0, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_8 \rangle;

    {\rm{(e)}}\ if \alpha_1 = 0, \alpha_2\neq0, then by choosing x = \sqrt[5]{\frac{\alpha_2}{\alpha_8}}, s = 0, t = 0, we have the family of representatives

    \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_8 \rangle;

    {\rm{(f)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[5]{\frac{\alpha_1}{\alpha_8}}, s = 0, t = 0, we have the family of representatives

    \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_8 \rangle.

    Summarizing all cases we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+ \alpha\nabla_2 +\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_8 \rangle^{ { \begin{array}{l}O(\alpha,\beta,\gamma,\mu) = O(\pm \alpha,\pm \eta_5^4 \beta,\eta_5^2\gamma,\eta_5^4\mu) = \\ O(\pm \alpha,\mp \eta_5^3 \beta,\eta_5^4\gamma,-\eta_5^3\mu) = O(\pm \alpha,\pm \eta_5^2 \beta,-\eta_5\gamma,\eta_5^2\mu) = \\ O(\pm \alpha,\mp \eta_5 \beta,-\eta_5^3\gamma,-\eta_5\mu) \end{array}}}, \\ \langle \nabla_1+\nabla_3 +\alpha\nabla_4 + \beta\nabla_6\rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(-\alpha,\beta) = \\ O(\alpha,-\beta) = O(-\alpha,-\beta) \end{array}}}, \\ \langle \nabla_1+ \alpha\nabla_3 +\beta \nabla_4+\nabla_7 \rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(-\eta_3\alpha,\eta_3^2\beta) = \\ O(\eta_3^2\alpha,-\eta_3\beta) \end{array}}}, \\ \langle \nabla_2+ \alpha\nabla_3+\nabla_5 + \beta\nabla_6+\gamma\nabla_7 \rangle ^{{ \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\alpha,\beta,-\gamma) = O(-\eta_3\alpha,\eta_3^2\beta,\gamma) = \\ O(\eta_3\alpha,\eta_3^2\beta,-\gamma) = O(\eta_3^2\alpha,-\eta_3\beta,\gamma) = O(-\eta_3^2\alpha,-\eta_3\beta,-\gamma) \end{array}}}, \\ \langle \nabla_2+ \alpha\nabla_4 +\beta\nabla_5 + \gamma\nabla_6+\nabla_8 \rangle ^{ { \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\alpha,\beta,\gamma) = O(\eta_5^4 \alpha,\eta_5^2\beta,\eta_5^4\gamma) = \\ O(-\eta_5^4\alpha,\eta_5^2\beta,\eta_5^4\gamma) = O(-\eta_5^3\alpha,\eta_5^4\beta,-\eta_5^3\gamma) = \\ O(\eta_5^3\alpha,\eta_5^4\beta,-\eta_5^3\gamma) = O(\eta_5^2\alpha,-\eta_5\beta,\eta_5^2\gamma) = \\ O(-\eta_5^2\alpha,-\eta_5\beta,\eta_5^2\gamma) = O(-\eta_5\alpha,-\eta_5^3\beta,-\eta_5\gamma) = \\ O(\eta_5\alpha,-\eta_5^3\beta,-\eta_5\gamma) \end{array}}}, \\ \langle \nabla_3 \rangle, \, \langle \nabla_3+ \nabla_4 +\alpha\nabla_6 \rangle, \, \langle \nabla_3+ \alpha\nabla_4+\nabla_7 \rangle ^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_3+ \nabla_5+\alpha\nabla_6+\beta\nabla_7 \rangle^{O(\alpha,\beta) = O(\alpha,-\beta)}, \, \langle \nabla_3+\nabla_6 \rangle, \\ \langle \nabla_4+ \alpha\nabla_5+\beta\nabla_6+\nabla_8 \rangle^{{ \begin{array}{l} O(\alpha,\beta) = O(-i\alpha,-\beta) = \\ O(i\alpha,-\beta) = O(-\alpha,\beta) \end{array}}}, \\ \langle \nabla_4+\nabla_7 \rangle, \, \langle \nabla_5+\nabla_6+ \alpha\nabla_7 \rangle^{O(\alpha,\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_5+\alpha\nabla_6+ \nabla_8 \rangle, \, \langle \nabla_5+ \alpha\nabla_7 \rangle^{O(\alpha) = O(-\alpha)}, \, \langle \nabla_6+ \nabla_8 \rangle, \, \langle \nabla_7 \rangle, \, \langle \nabla_8 \rangle,\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{150}^{\alpha, \beta, \gamma, \mu } & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{151}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4+\beta e_5 \\ {\mathbf{N}}_{152}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{153}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = e_4+\beta e_5 & e_3e_4 = \gamma e_5 \\ {\mathbf{N}}_{154}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4+\gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{155} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_3e_3 = e_4 \\ {\mathbf{N}}_{156}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = e_5 & e_3e_3 = e_4+\alpha e_5 \\ {\mathbf{N}}_{157}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{158}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = e_4+\alpha e_5 & e_3e_4 = \beta e_5 \\ {\mathbf{N}}_{159} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_3e_3 = e_4+e_5 \\ {\mathbf{N}}_{160}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4+\beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{161} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 \\&& e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{162}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\& & e_3e_3 = e_4+e_5 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{163}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\&& e_3e_3 = e_4+\alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{164}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\&& e_3e_3 = e_4 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{165} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4+e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{166} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{167} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{13}^{4*}(\lambda):

    \begin{array}{|l|llll|} \hline {\mathbf{N}}^{4*}_{13}(\lambda) & \;\; e_1e_1 = e_2 \;\; e_1e_2 = e_3 \;\; e_1e_3 = e_4 \;\; e_2e_2 = \lambda e_4 \\ \hline & { \begin{array}{l} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(2)) = \langle [\Delta_{22}],4[\Delta_{23}]+[\Delta_{14}],[\Delta_{24}]\rangle,\\\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{13}(2)) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(2))\oplus \langle [\Delta_{23}], [\Delta_{33}], [\Delta_{34}], [\Delta_{44}]\rangle \\\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(\lambda )_{\lambda \neq2}) = \langle [\Delta_{22}],(3\lambda-2)[\Delta_{23}]+[\Delta_{14}]\rangle,\\\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{13}(\lambda)_{\lambda \neq2}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{13}(\lambda)\oplus \langle [\Delta_{23}], [\Delta_{24}], [\Delta_{33}], [\Delta_{34}], [\Delta_{44}] \rangle \end{array} }\\ \hline &{ \phi = \begin{pmatrix} x&0&0&0\\ y&x^2&0&0\\ z&2xy&x^3&0\\ t&\lambda y^2+2xz&(\lambda+2)x^2y&x^4 \end{pmatrix} }\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{lllll} \nabla_1 = [\Delta_{14}]+(3\lambda-2)[\Delta_{23}], & \nabla_2 = [\Delta_{22}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}({\mathbf N}_{13}^{4*}(\lambda)) . Since

    \phi^T\begin{pmatrix} 0&0&0&\alpha_1\\ 0&\alpha_2&(3\lambda-2)\alpha_1+\alpha_3&\alpha_4\\ 0&(3\lambda-2)\alpha_1+\alpha_3&\alpha_5&\alpha_6\\ \alpha_1&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^{**}&\alpha^{***}&\alpha^{*}&\alpha_1^*\\ \alpha^{***}&\alpha_2^*+\lambda\alpha^{*}&(3\lambda-2)\alpha^*_1+\alpha_3^*&\alpha^*_4\\ \alpha^{*}&(3\lambda-2)\alpha^*_1+\alpha_3^*&\alpha^*_5&\alpha^*_6\\ \alpha^*_1&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix},

    we have

    \begin{array}{lcl} \alpha_1^* = (\alpha_1x+\alpha_4y+\alpha_6z+\alpha_7t)x^4, \\ \alpha_2^* = \alpha_2x^4+4\lambda(\alpha_6y+\alpha_7z)xy^2+\lambda^2\alpha_7y^4+4(\alpha_4z+(\alpha_3+(3\lambda-2)\alpha_1)y)x^3 \\ \qquad\qquad{ +2(4\alpha_6yz+2\alpha_7z^2+(2\alpha_5+\lambda\alpha_4)y^2)x^2 }\\ \qquad{ -\lambda((\lambda+2)(\alpha_4y+\alpha_6z+\alpha_7t)y+((\alpha_3+4\lambda\alpha_1)y+\alpha_5z+\alpha_6t)x)x^2, }\\ \alpha_3^* = [(\lambda+2)(\alpha_4x^2+2\alpha_6xy+2\alpha_7xz+\lambda\alpha_7y^2)y \\\qquad { +((\alpha_3+(3\lambda-2)\alpha_1)x^2+2\alpha_5xy+2\alpha_6xz+\lambda\alpha_6y^2)x]x^2 }\\\qquad\qquad\qquad { -(3\lambda-2)(\alpha_1x+\alpha_4y+\alpha_6z+\alpha_7t)x^4, }\\ \alpha_4^* = (\alpha_4x^2+2\alpha_6xy+2\alpha_7xz+\lambda\alpha_7y^2)x^4, \\ \alpha_5^* = (\alpha_5x^2+2(\lambda+2)\alpha_6xy+(\lambda+2)^2\alpha_7y^2)x^4, \\ \alpha_6^* = (\alpha_6x+(\lambda+2)\alpha_7y)x^6, \\ \alpha_7^* = \alpha_7x^8. \end{array}

    We are interested in

    (\alpha_3,\alpha_4,\alpha_5,\alpha_6,\alpha_7)\neq(0,0,0,0,0) {\text{ and }}(\alpha_1,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0).

    Let us consider the following cases:

    1.\ \alpha_7 = 0, \alpha_6 = 0, \alpha_5 = 0, \alpha_4 = 0, then \alpha_1\neq0, \alpha_3\neq0 and

    {\rm{(a)}}\ if \lambda\notin \{ 1,2,4\}, (\lambda-4)\alpha_3\neq4(1-\lambda)(\lambda-2)\alpha_1, then by choosing y = \frac{\alpha_2x}{(\lambda-4)\alpha_3+4(\lambda-1)(\lambda-2)\alpha_1}, we have the family of representatives

    \langle \alpha\nabla_1+\nabla_3 \rangle_{\alpha\notin \Big\{ 0,\frac{(\lambda-4)}{4(1-\lambda)(\lambda-2) }\Big\};\, \lambda\neq 1,2,4};

    {\rm{(b)}}\ if \lambda\notin \{ 1,2,4\}, (\lambda-4)\alpha_3 = 4(1-\lambda)(\lambda-2)\alpha_1, \alpha_2 = 0, then we have the family of representatives \langle \frac{\lambda-4}{4(1-\lambda)(\lambda-2)}\nabla_1+\nabla_3 \rangle_{\lambda\neq 1,2,4}, which we will be jointed with the family from the case (1a);

    {\rm{(c)}}\ if \lambda\notin \{ 1,2,4\}, (\lambda-4)\alpha_3 = 4(1-\lambda)(\lambda-2)\alpha_1, \alpha_2\neq0, then by choosing x = \frac{\alpha_2}{\alpha_3}, y = 0, z = 0, t = 0, we have the family of representatives \langle (\lambda-4)\nabla_1+4(1-\lambda)(\lambda-2) (\nabla_2+\nabla_3) \rangle_{\lambda\neq 1,2,4};

    {\rm{(d)}}\ if \lambda\in \{ 1,2,4\}, then by choosing some suitable x and y we have the family of representatives \langle \alpha\nabla_1+\nabla_3 \rangle_{\alpha\neq 0, \lambda\in \{1,2,4\}}, which will be jointed with the family from the case (1a).

    2. \alpha_7 = 0, \alpha_6 = 0, \alpha_5 = 0, \alpha_4\neq0, then we have

    {\rm{(a)}}\ if \alpha_3 = 2(2-\lambda)\alpha_1 , then by choosing

    x = 4 \alpha_4^2, y = -4 \alpha_1 \alpha_4, z = \alpha_1 \alpha_3 (4-\lambda)-\alpha_2 \alpha_4-\alpha_1^2 (8-12 \lambda+3 \lambda^2), t = 0,

    we have the representative \langle \nabla_4 \rangle;

    {\rm{(b)}}\ if \alpha_3\neq2(2-\lambda)\alpha_1 , then by choosing

    \begin{array}{c} x = \frac{\alpha_3+2(\lambda-2)\alpha_1}{\alpha_4}, y = -\frac{\alpha_1(\alpha_3+2(\lambda-2)\alpha_1)}{\alpha_4^2}, \\z = \frac{(2(2-\lambda)\alpha_1-\alpha_3)(\alpha_2\alpha_4+(\lambda-4)\alpha_1\alpha_3+(3\lambda^2-12\lambda+8)\alpha_1^2)}{4\alpha^3_4}, t = 0,\end{array}

    we have the representative \langle \nabla_3+\nabla_4 \rangle .

    3. \alpha_7 = 0, \alpha_6 = 0, \alpha_5\neq0, then

    {\rm{(a)}}\ if \alpha_4 = 0, then \alpha_1\neq0 and

    {\rm{(i)}}\ if \lambda\neq0, then by choosing

    x = \frac{\alpha_1}{\alpha_5}, y = -\frac{\alpha_1\alpha_3}{2\alpha_5^2}, z = \frac{\alpha_1(2\alpha_2\alpha_5+(\lambda-2)\alpha_3^2+4(\lambda^2-3\lambda+2)\alpha_1\alpha_3)}{2\lambda\alpha_5^3}, t = 0,

    we have the family of representatives \langle \nabla_1+\nabla_5 \rangle_{\lambda\neq0};

    {\rm{(ii)}}\ if \lambda = 0, then by choosing x = \frac{\alpha_1}{\alpha_5}, y = -\frac{\alpha_1\alpha_3}{2\alpha_5^2}, z = 0, t = 0, we have the family representative \langle \nabla_1+\alpha\nabla_2+\nabla_5 \rangle_{\alpha\neq0,\lambda = 0} and the representative \langle \nabla_1+\nabla_5 \rangle_{\lambda = 0}, which will be jointed with the family from the case (3(a)i).

    (b) if \alpha_4\neq0 and \lambda = 0, then we have the followings:

    {\rm{(i)}}\ if \alpha_3\alpha_4 = 2\alpha_1(2\alpha_4+\alpha_5), then by choosing

    x = 4 \alpha_4^3, y = -4 \alpha_1 \alpha_4^2, z = 4 \alpha_1 \alpha_3 \alpha_4-\alpha_2 \alpha_4^2-4 \alpha_1^2 (2 \alpha_4+\alpha_5), t = 0,

    we have the family of representatives \langle \alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq0,\lambda = 0};

    {\rm{(ii)}}\ if \alpha_3\alpha_4\neq 2\alpha_1(2\alpha_4+\alpha_5), then by choosing

    \begin{array}{c} x = \frac{\alpha_3\alpha_4-2\alpha_1(2\alpha_4+\alpha_5)}{\alpha_4\alpha_5)}, y = \frac{\alpha_1(2\alpha_1(2\alpha_4+\alpha_5)-\alpha_3\alpha_4)}{\alpha^2_4\alpha_5)},\\z = \frac{(2\alpha_1(2\alpha_4+\alpha_5)-\alpha_3\alpha_4)(\alpha_2\alpha_4^2-4\alpha_1\alpha_3\alpha_4+4\alpha_1^2(2\alpha_4+\alpha_5))}{4\alpha_4^4\alpha_5}, t = 0, \end{array}

    we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq0, \lambda = 0} ;

    (c) if \alpha_4\neq0 and \lambda\neq 0, then we have the followings:

    {\rm{(i)}}\ if 4\alpha_4 = \lambda\alpha_5, 4\lambda(\lambda-4)\alpha_1\alpha_3+\lambda^2\alpha_2\alpha_5+4(3\lambda^3-12\lambda^2+8\lambda+16)\alpha_1^2 = 0, \lambda\alpha_3+2(\lambda^2-2\lambda-4)\alpha_1 = 0, then by choosing x = 1, y = -\frac{\alpha_1}{\alpha_4}, z = 0, t = 0, we have the family of representatives \langle \frac{\lambda}{4}\nabla_4+\nabla_5 \rangle_{\lambda\neq0} ;

    {\rm{(ii)}}\ if 4\alpha_4 = \lambda\alpha_5, 4\lambda(\lambda-4)\alpha_1\alpha_3+\lambda^2\alpha_2\alpha_5+4(3\lambda^3-12\lambda^2+8\lambda+16)\alpha_1^2 = 0, \lambda\alpha_3+2(\lambda^2-2\lambda-4)\alpha_1\neq0, then by choosing

    x = \frac{\lambda\alpha_3+2(\lambda^2-2\lambda-4)\alpha_1}{\lambda\alpha_5}, y = -\frac{4\alpha_1(\lambda\alpha_3+2(\lambda^2-2\lambda-4)\alpha_1)}{\lambda^2\alpha^2_5}, z = 0, t = 0,

    we have the family of representatives \langle \nabla_3+\frac{\lambda}{4}\nabla_4+\nabla_5 \rangle_{\lambda\neq0} ;

    {\rm{(iii)}}\ if 4\alpha_4 = \lambda\alpha_5, 4\lambda(\lambda-4)\alpha_1\alpha_3+\lambda^2\alpha_2\alpha_5+4(3\lambda^3-12\lambda^2+8\lambda+16)\alpha_1^2\neq0, then by choosing

    \begin{array}{c} x = \frac{\sqrt{4\lambda(\lambda-4)\alpha_1\alpha_3+\lambda^2\alpha_2\alpha_5+4(3\lambda^3-12\lambda^2+8\lambda+16)\alpha_1^2}}{\lambda\alpha_5}, \\ y = -\frac{4\alpha_1\sqrt{4\lambda(\lambda-4)\alpha_1\alpha_3+\lambda^2\alpha_2\alpha_5+4(3\lambda^3-12\lambda^2+8\lambda+16)\alpha_1^2}}{\lambda^2\alpha^2_5}, z = 0, t = 0,\end{array}

    we have the family of representatives \langle \nabla_2+\alpha\nabla_3+\frac{\lambda}{4}\nabla_4+\nabla_5 \rangle_{\lambda\neq0} ;

    {\rm{(iv)}}\ if \lambda\neq0, 4\alpha_4\neq\lambda\alpha_5 , then by choosing

    y = -\frac{\alpha_1}{\alpha_4}x, z = -\frac{\alpha_2\alpha_4^2+(\lambda-4)\alpha_1\alpha_3\alpha_4+\alpha_1^2(4\alpha_5+(3\lambda^2-12\lambda+8)\alpha_4)}{\alpha^2_4(4\alpha_4-\lambda\alpha_5)}x, t = 0,

    we have two families of representatives

    \langle \alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq\frac{\lambda}{4}} {\text{ and }} \langle \nabla_3+\alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq\frac{\lambda}{4}}

    depending on \alpha_3\alpha_4-2\alpha_1\alpha_5+2(\lambda-2)\alpha_1\alpha_4 = 0 or not. These families will be jointed with representatives from cases (3(c)i) and (3(c)ii).

    4. \alpha_7 = 0, \alpha_6\neq0, then by choosing y = -\frac{\alpha_4}{2\alpha_6}x, z = -\frac{\alpha_4^2-2\alpha_1\alpha_6}{2\alpha^2_6}x, we have \alpha_1^* = 0, \alpha_4^* = 0. Since we can suppose that \alpha_1 = 0, \alpha_4 = 0 and

    {\rm{(a)}}\ if \lambda\neq0, \alpha_3 = 0, then by choosing t = {\frac{\alpha_2}{\lambda\alpha_6}}x, we have the representatives \langle \nabla_6 \rangle _{\lambda\neq0} and \langle \nabla_5+\nabla_6 \rangle _{\lambda\neq0} depending on \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \lambda\neq0, \alpha_3\neq0, then by choosing x = \sqrt{\frac{\alpha_3}{\alpha_6}}, t = {\frac{\alpha_2\sqrt{\alpha_3}}{\lambda\alpha_6\sqrt{\alpha_6}}}, we have the family of representatives \langle \nabla_3+\alpha\nabla_5+\nabla_6 \rangle _{\lambda\neq0};

    {\rm{(c)}}\ if \lambda = 0, \alpha_2 = 0, \alpha_3 = 0, then we have the representatives \langle \nabla_6 \rangle _{\lambda = 0} and \langle \nabla_5+\nabla_6 \rangle _{\lambda = 0} depending on \alpha_5 = 0 or not, which will be jointed with representatives from the case (4a);

    {\rm{(d)}}\ if \lambda = 0, \alpha_2 = 0, \alpha_3\neq0, then by choosing x = \sqrt{\frac{\alpha_3}{\alpha_6}}, t = 0 , we have the family of representatives \langle \nabla_3+\alpha\nabla_5+\nabla_6 \rangle _{(\lambda = 0)}, which will be jointed with the family of representatives from the case (4b);

    {\rm{(e)}}\ if \lambda = 0, \alpha_2\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_2}{\alpha_6}}, t = 0, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5+\nabla_6 \rangle _{\lambda = 0}.

    5. \alpha_7\neq0, \lambda\neq-2, then by choosing

    \begin{array}{c} y = -\frac{\alpha_6}{\alpha_7(\lambda+2)}x, z = \frac{2(\lambda+2)^2\alpha_4\alpha_7-(\lambda+4)\alpha_6^2}{2(\lambda+2)^2\alpha^2_7}x,\\ t = \frac{(\lambda^2+6\lambda+8)\alpha_4\alpha_6\alpha_7-2(\lambda+2)^2\alpha_1\alpha^2_7-(\lambda+4)\alpha^3_6}{2(\lambda+2)^2\alpha^3_7}x ,\end{array}

    we have \alpha_1^* = 0, \alpha_4^* = 0, \alpha_6^* = 0. Now we can suppose that \alpha_1 = 0, \alpha_4 = 0, \alpha_6 = 0 then we have

    {\rm{(a)}}\ if \alpha_3 = 0, \alpha_5 = 0, \alpha_2 = 0, then we have the representative \langle \nabla_7 \rangle_{\lambda\neq-2} ;

    {\rm{(b)}}\ if \alpha_3 = 0, \alpha_5 = 0, \alpha_2\neq0, then by choosing x = \sqrt[4]{\alpha_2\alpha_7^{-1}}, we have the representative \langle \nabla_2+\nabla_7 \rangle_{\lambda\neq-2} ;

    {\rm{(c)}}\ if \alpha_5\neq0, then by choosing x = \sqrt{\alpha_5\alpha_7^{-1}}, y = -\frac{\alpha_3}{2 \sqrt{\alpha_5 \alpha_7}}, z = \frac{\alpha_3^2}{ 4 \sqrt{\alpha_5^3\alpha_7}}, t = 0, we have the family of representatives \langle \alpha\nabla_2+\nabla_5+\nabla_7 \rangle_{\lambda\neq-2} .

    {\rm{(d)}}\ if \alpha_3\neq0, \alpha_5 = 0 then by choosing x = \sqrt[3]{\alpha_3\alpha_7^{-1}}, we have the family of representatives \langle \alpha\nabla_2+\nabla_3+ \nabla_7 \rangle_{\lambda\neq-2} .

    6. \alpha_7\neq0, \lambda = -2, then

    {\rm{(a)}}\ if \alpha_6 = 0, \alpha_5 = 0, then by choosing z = \frac{y^2}{x}-\frac{\alpha_4x}{2\alpha_7}, t = -\frac{x \alpha_1+y \alpha_4}{\alpha_7}, we have \alpha_1^* = 0 and \alpha_4^* = 0. Now consider the followings:

    {\rm{(i)}}\ if \alpha_3 = 8\alpha_1, \alpha_2\alpha_7-\alpha_4^2 = 0, then we have the representative \langle \nabla_7 \rangle_{\lambda = -2}, which will be jointed with the representative from the case (5a);

    {\rm{(ii)}}\ if \alpha_3 = 8\alpha_1, \alpha_2\alpha_7-\alpha_4^2\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_2\alpha_7-\alpha_4^2}{\alpha_7^2}}, y = 0, we have the representative \langle \nabla_2+\nabla_7 \rangle_{\lambda = -2}, which will be jointed with the representative from the case (5b);

    {\rm{(iii)}}\ if \alpha_3\neq8\alpha_1, then by choosing x = \sqrt[3]{\frac{\alpha_3-8\alpha_1}{\alpha_7}}, y = \frac{\alpha_2\alpha_7-\alpha_4^2}{48\alpha_1\alpha_7-6\alpha_3\alpha_7}x, we have the representative \langle \nabla_3+\nabla_7 \rangle_{\lambda = -2} .

    (b) if \alpha_6 = 0, \alpha_5\neq0, then by choosing

    x = \sqrt{\frac{\alpha_5}{\alpha_7}}, y = \frac{8\alpha_1-\alpha_3}{2\sqrt{\alpha_5\alpha_7}}, z = \frac{\alpha_7(\alpha_3-8\alpha_1)^2-2\alpha_4\alpha_5^2}{4\alpha_5\alpha_7\sqrt{\alpha_5\alpha_7}}, t = \frac{\alpha_3\alpha_4-2\alpha_1(4\alpha_4+\alpha_5)}{2\alpha_7\sqrt{\alpha_5\alpha_7}},

    we have the family of representatives \langle \alpha\nabla_2+\nabla_5+\nabla_7 \rangle_{\lambda = -2}, which will be jointed with the representative from the case (5c).

    (c) if \alpha_6\neq0, then we have the following cases:

    {\rm{(i)}}\ if \alpha_5\alpha_7 = \alpha_6^2, 8\alpha_1\alpha_7+\alpha_4\alpha_6 = \alpha_3\alpha_7, then by choosing

    x = \frac{\alpha_6}{\alpha_7}, y = 0, z = -\frac{\alpha_4\alpha_6}{2\alpha_7^2}, t = \frac{\alpha_6(\alpha_4\alpha_6-2\alpha_1\alpha_7)}{2\alpha_7^3},

    we have the family of representatives \langle \alpha\nabla_2+\nabla_5+\nabla_6+\nabla_7 \rangle_{\lambda = -2};

    {\rm{(ii)}}\ if \alpha_5\alpha_7 = \alpha_6^2, 8\alpha_1\alpha_7+\alpha_4\alpha_6\neq \alpha_3\alpha_7, then by choosing

    \begin{array}{c} x = \frac{\alpha_6}{\alpha_7}, y = \frac{\alpha_6 (\alpha_2 \alpha_7-\alpha_4^2 - 2 \alpha_1 \alpha_6 )}{ 6 \alpha_7 (\alpha_4 \alpha_6 + 8 \alpha_1 \alpha_7 - \alpha_3 \alpha_7))},\\ z = \frac{y^2}{x}-\frac{\alpha_4x}{2\alpha_7}-\frac{\alpha_6y}{\alpha_7}, t = -\frac{x \alpha_1 + y \alpha_4 + z \alpha_6}{\alpha_7},\end{array}

    we have the family of representatives

    \langle \alpha\nabla_3+\nabla_5+\nabla_6+\nabla_7 \rangle_{\alpha\neq0,\lambda = -2};

    {\rm{(iii)}}\ if \alpha_5\alpha_7-\alpha_6^2\neq0, then by choosing

    \begin{array}{c} x = \frac{\alpha_6}{\alpha_7}, y = \frac{\alpha_6 (\alpha_4 \alpha_6+8 \alpha_1 \alpha_7-\alpha_3 \alpha_7)}{2 \alpha_7 (-\alpha_6^2+\alpha_5 \alpha_7)}, z = \frac{y^2}{x}-\frac{\alpha_4}{2\alpha_7}x-\frac{\alpha_6}{\alpha_7}y,\\t = \frac{(\alpha_4\alpha_6-2\alpha_1\alpha_7)x^2-2\alpha_6\alpha_7y^2+2(\alpha_6^2-\alpha_4\alpha_7)xy}{2\alpha_7^2x},\end{array}

    we have the family of representatives

    \langle \alpha\nabla_2+\beta\nabla_5+\nabla_6+\nabla_7 \rangle_{\beta\neq1,\lambda = -2},

    which will be jointed with the family from the case (6(c)i).

    Summarizing all cases we have the following distinct orbits:

    \begin{array}{c} \langle (\lambda-4)\nabla_1+4(1-\lambda)(\lambda-2)(\nabla_2+\nabla_3) \rangle_{\lambda \notin \{ 1; 2; 4 \}}, \langle \nabla_1+\alpha\nabla_2+\nabla_5 \rangle_{\lambda = 0, \alpha \neq 0}, \\ \langle \alpha\nabla_1+\nabla_3 \rangle_{\alpha\neq0}, \, \langle \nabla_1+\nabla_5 \rangle, \langle \nabla_2+\alpha\nabla_3+\frac{\lambda}{4}\nabla_4+\nabla_5 \rangle_{\lambda\neq0}^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5+\nabla_6 \rangle _{\lambda = 0}^{O(\alpha,\beta) = O(\eta_3^2\alpha,-\eta\beta) = O(-\eta_3\alpha,\eta_3^2\beta)}, \\ \langle \alpha\nabla_2+\nabla_3+ \nabla_7 \rangle_{\lambda\neq-2}^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \alpha\nabla_2+\beta\nabla_5+\nabla_6+\nabla_7 \rangle_{\lambda = -2}, \\ \langle \alpha\nabla_2+\nabla_5+\nabla_7 \rangle, \langle \nabla_2 + \nabla_7 \rangle, \langle \nabla_3+\nabla_4 \rangle, \langle \nabla_3+\alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq0}, \\ \langle \nabla_3+\alpha\nabla_5+\nabla_6 \rangle, \langle \alpha\nabla_3+\nabla_5+\nabla_6+\nabla_7 \rangle_{\alpha\neq0,\lambda = -2}, \langle \nabla_3+\nabla_7 \rangle_{\lambda = -2}, \langle \nabla_4 \rangle_{\lambda\neq 2}, \\ \langle \alpha\nabla_4+\nabla_5 \rangle_{\alpha\neq0}, \langle \nabla_5+\nabla_6 \rangle, \langle \nabla_6 \rangle, \langle \nabla_7 \rangle. \end{array}

    Now we have the following new algebras

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{168}^{\lambda \neq 1; 2; 4} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = (\lambda-4)e_5 \\ & & { e_2e_2 = \lambda e_4 + 4(1-\lambda)(\lambda-2)e_5 } & { e_2e_3 = - \lambda(\lambda+2)e_5 }\\ {\mathbf{N}}_{169}^{\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = -2 e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{170}^{\lambda, \alpha \neq 0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_2e_2 = \lambda e_4 & { e_2e_3 = (1+\alpha(3\lambda-2)) e_5 } \\ {\mathbf{N}}_{171}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \lambda e_4 & { e_2e_3 = (3\lambda-2) e_5 } & e_3e_3 = e_5 \\ {\mathbf{N}}_{172}^{\lambda \neq 0,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 + e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = \frac{\lambda} {4} e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{173}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{174}^{\lambda\neq-2, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 + \alpha e_5 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{175}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = -2 e_4+\alpha e_5 \\ && e_3e_3 = \beta e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{176}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4+\alpha e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{177}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4+ e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{178}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 \\ {\mathbf{N}}_{179}^{\lambda,\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{180}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{181}^{\alpha \neq 0 } & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = -2 e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{182} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = -2 e_4 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{183}^{\lambda\neq 2} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = \lambda e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{184}^{\lambda, \alpha\neq 0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{185}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = \lambda e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{186}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{187}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = \lambda e_4 & e_4e_4 = e_5 \end{array}

    Here we will collect all information about {\mathbf N}_{14}^{4*}:

    \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{14} & \begin{array}{l} e_1e_2 = e_3 \\ e_1e_3 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{14})& = &\\ {\langle [\Delta_{11}],[\Delta_{22}],[\Delta_{23}],[\Delta_{33}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{14})& = &\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{14})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}]\rangle } \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&q&0&0\\ 0&r&xq&0\\ t&s&xr&x^2q \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{8}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{14}) . Since

    \phi^T\begin{pmatrix} \alpha_1&0&0&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ 0&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{**}&\alpha_2^*\\ \alpha^{*}&\alpha_3^*&\alpha^*_4&\alpha^*_5\\ \alpha^{**}&\alpha^*_4&\alpha^*_6&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix},

    we have

    \begin{array}{lcl} \alpha_1^* & = & \alpha_1x^2+2\alpha_2xt+\alpha_8t^2, \\ \alpha_2^* & = & (\alpha_2x+\alpha_8t)x^2q, \\ \alpha_3^* & = & (\alpha_3q+\alpha_4r+\alpha_5s)q+(\alpha_4q+\alpha_6r+\alpha_7s)r+(\alpha_5q+\alpha_7r+\alpha_8s)s, \\ \alpha_4^* & = & (\alpha_4q+\alpha_6r+\alpha_7s)xq+(\alpha_5q+\alpha_7r+\alpha_8s)xr, \\ \alpha_5^* & = & (\alpha_5q+\alpha_7r+\alpha_8s)x^2q, \\ \alpha_6^* & = & (\alpha_6q^2+2\alpha_7qr+\alpha_8r^2)x^2, \\ \alpha_7^* & = & (\alpha_7q+\alpha_8r)x^3q, \\ \alpha_8^* & = & \alpha_8x^4q^2. \end{array}

    We are interested in (\alpha_2,\alpha_5,\alpha_7,\alpha_8)\neq(0,0,0,0) . Let us consider the following cases:

    1.\ \alpha_8 = 0, \alpha_7 = 0, \alpha_5 = 0, then \alpha_2\neq0 and we have

    {\rm{(a)}}\ if \alpha_6 = 0, \alpha_4 = 0, \alpha_3 = 0, then by choosing x = 2\alpha_2, q = 1, r = 0, s = 0, t = -\alpha_1, we have the representative \langle \nabla_2 \rangle;

    {\rm{(b)}}\ if \alpha_6 = 0, \alpha_4 = 0, \alpha_3\neq0, then by choosing x = \alpha_3, q = \alpha_2\alpha_3^2, r = 0, s = 0, t = -\frac{\alpha_1\alpha_3}{2\alpha_2}, we have the representative \langle \nabla_2+\nabla_3 \rangle;

    {\rm{(c)}}\ if \alpha_6 = 0, \alpha_4\neq0, then by choosing x = \alpha_4, q = \alpha_2\alpha_4, r = -\frac{\alpha_2\alpha_3}{2}, s = 0, t = -\frac{\alpha_1\alpha_4}{2\alpha_2}, we have the representative \langle \nabla_2+\nabla_4 \rangle;

    {\rm{(d)}}\ if \alpha_6\neq0, \alpha_3\alpha_6-\alpha_4^2 = 0, then by choosing x = \alpha_6, q = \alpha_2, r = -\frac{\alpha_2\alpha_4}{\alpha_6}, s = 0, t = -\frac{\alpha_1\alpha_6}{2\alpha_2}, we have the representative \langle \nabla_2+\nabla_6 \rangle;

    {\rm{(e)}}\ if \alpha_6\neq0, \alpha_3\alpha_6-\alpha_4^2\neq0, then by choosing

    \begin{array}{c} x = \frac{\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{\alpha_6},q = \frac{\alpha_2\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{\alpha^2_6},r = -\frac{\alpha_2\alpha_4\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{\alpha^3_6},\\ s = 0, t = -\frac{\alpha_1\sqrt{\alpha_3\alpha_6-\alpha_4^2}}{2\alpha_2\alpha_6}, \end{array}

    we have the representative \langle \nabla_2+\nabla_3+\nabla_6 \rangle.

    2. \alpha_8 = 0, \alpha_7 = 0, \alpha_5\neq0, then we have

    {\rm{(a)}}\ if \alpha_6 = 0, \alpha_2 = 0, then by choosing

    x = 1, r = -\frac{\alpha_4}{\alpha_5}q, s = \frac{2\alpha_4^2-\alpha_3\alpha_5}{2\alpha_5^2}q, t = 0,

    we have the representatives \langle \nabla_5 \rangle and \langle \nabla_1+\nabla_5 \rangle depending on whether \alpha_1 = 0 or not;

    {\rm{(b)}}\ if \alpha_6 = 0, \alpha_2\neq0, then by choosing

    x = \alpha_5, q = \alpha_2, r = -\frac{\alpha_2\alpha_4}{\alpha_5}, s = \frac{\alpha_2(2\alpha_4^2-\alpha_3\alpha_5)}{2\alpha_5^2}, t = -\frac{\alpha_1\alpha_5}{2\alpha_2},

    we have the representatives \langle \nabla_2+\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_6\neq0, \alpha_5 = -\alpha_6, then we have the following subcases:

    {\rm{(i)}}\ if \alpha_2 = 0, \alpha_4 = 0, \alpha_1 = 0, then we have the representative \langle \nabla_5-\nabla_6 \rangle;

    {\rm{(ii)}}\ if \alpha_2 = 0, \alpha_4 = 0, \alpha_1\neq0, then by choosing

    x = 1, q = \sqrt{\frac{\alpha_1}{\alpha_5}}, r = 0, s = -\frac{\alpha_3\sqrt{\alpha_1}}{2\alpha_5\sqrt{\alpha_5}}, t = 0,

    we have the representative \langle \nabla_1+\nabla_5-\nabla_6 \rangle;

    {\rm{(iii)}}\ if \alpha_2 = 0, \alpha_4\neq0, \alpha_1 = 0, then by choosing

    x = {\frac{\alpha_4}{\alpha_5}}, q = 1, r = 0, s = -\frac{\alpha_3}{2\alpha_5}, t = 0,

    we have the representative \langle \nabla_4+\nabla_5-\nabla_6 \rangle;

    {\rm{(iv)}}\ if \alpha_2 = 0, \alpha_4\neq0, \alpha_1\neq0, then by choosing

    x = \frac{\alpha_4}{\alpha_5}, q = \sqrt{\frac{\alpha_1}{\alpha_5}}, r = 0, s = -\frac{\alpha_3\sqrt{\alpha_1}}{2\alpha_5\sqrt{\alpha_5}}, t = 0,

    we have the representative \langle \nabla_1+\nabla_4+\nabla_5-\nabla_6 \rangle;

    {\rm{(v)}}\ if \alpha_2\neq0, \alpha_4 = 0, then by choosing

    x = \alpha_5, q = \alpha_2, r = 0, s = -\frac{\alpha_2\alpha_3}{2\alpha_5}, t = -\frac{\alpha_1\alpha_5}{2\alpha_2},

    we have the representative \langle \nabla_2+\nabla_5-\nabla_6 \rangle;

    {\rm{(vi)}}\ if \alpha_2\neq0, \alpha_4\neq0, then by choosing

    x = \frac{\alpha_4}{\alpha_5}, q = {\frac{\alpha_2\alpha_4}{\alpha^2_5}}, r = 0, s = -\frac{\alpha_2\alpha_3\alpha_4}{2\alpha_5^3}, t = -\frac{\alpha_1\alpha_4}{2\alpha_2\alpha_5},

    we have the representative \langle \nabla_2+\nabla_4+\nabla_5-\nabla_6 \rangle.

    (d) if \alpha_6\neq0, \alpha_5\neq-\alpha_6, then we have the following subcases:

    {\rm{(i)}}\ if \alpha_2 = 0, \alpha_1 = 0, then by choosing

    x = 1, q = 1, s = \frac{ \alpha_4^2 (2 \alpha_5+\alpha_6)-\alpha_3 (\alpha_5+\alpha_6)^2}{2 \alpha_5 (\alpha_5+\alpha_6)^2}, r = -\frac{\alpha_4}{\alpha_5+\alpha_6}, t = 0,

    we have the family of representatives \langle \nabla_5+\alpha\nabla_6 \rangle_{\alpha\neq0,-1}, which will be jointed with the cases (2a) and (2(c)i);

    {\rm{(ii)}}\ if \alpha_2 = 0, \alpha_1\neq0, then by choosing

    \begin{array}{c} x = 1, q = \sqrt{\frac{\alpha_1}{\alpha_5}}, r = -\frac{\alpha_4\sqrt{\alpha_1}}{(\alpha_5+\alpha_6)\sqrt{\alpha_5}}, \\s = \frac{(\alpha_4^2(2\alpha_5+\alpha_6)-\alpha_3(\alpha_5+\alpha_6)^2)\sqrt{\alpha_1}}{2\alpha_5(\alpha_5+\alpha_6)^2\sqrt{\alpha_5}}, t = 0, \end{array}

    we have the family of representatives \langle \nabla_1+\nabla_5+\alpha\nabla_6 \rangle_{\alpha\neq0,-1}, which will be jointed with the cases (2a) and (2(c)ii);

    {\rm{(iii)}}\ if \alpha_2\neq0, then by choosing

    x = \alpha_5, q = \alpha_2, r = -\frac{\alpha_2\alpha_4}{\alpha_5+\alpha_6}, s = \frac{\alpha_2(\alpha_4^2(2\alpha_5+\alpha_6)-\alpha_3(\alpha_5+\alpha_6)^2)}{2\alpha_5(\alpha_5+\alpha_6)^2}, t = -\frac{\alpha_1\alpha_5}{2\alpha_2},

    we have the family of representatives \langle \nabla_2+\nabla_5+\alpha\nabla_6 \rangle_{\alpha\neq0,-1}, which will be jointed with the cases (2b) and (2(c)v).

    3. \alpha_8 = 0, \alpha_7\neq0 then by choosing r = -\frac{\alpha_5}{\alpha_7}q, s = \frac{\alpha_5\alpha_6-\alpha_4\alpha_7}{\alpha_7^2}q, we have \alpha_4^* = \alpha_5^* = 0. Therefore, we can suppose that \alpha_4 = 0, \alpha_5 = 0, thus we have

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_1 = 0, \alpha_3 = 0, then we have the representatives \langle \nabla_7 \rangle and \langle \nabla_6+\nabla_7\rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_3}{\alpha_7}}, q = 1, r = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_3+\alpha\nabla_6+\nabla_7\rangle;

    {\rm{(c)}}\ if \alpha_2 = 0, \alpha_1\neq0, \alpha_3 = 0, then we have the representatives \langle \nabla_1+\nabla_7\rangle and \langle \nabla_1+\nabla_6+\nabla_7\rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(d)}}\ if \alpha_2 = 0, \alpha_1\neq0, \alpha_3\neq0, then by choosing

    x = \sqrt[3]{\frac{\alpha_3}{\alpha_7}}, q = \sqrt[6]{\frac{\alpha^3_1}{\alpha_3\alpha^2_7}}, r = 0, s = 0, t = 0,

    we have the family of representatives \langle \nabla_1+\nabla_3+\alpha\nabla_6+\nabla_7\rangle;

    {\rm{(e)}}\ if \alpha_2\neq0, \alpha_3 = 0, then by choosing q = \frac{\alpha_2}{\alpha_7}, r = 0, s = 0, t = -\frac{\alpha_1}{2\alpha_2}x, we have the representatives \langle \nabla_2+\nabla_7 \rangle and \langle \nabla_2+\nabla_6+\nabla_7\rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(f)}}\ if \alpha_2\neq0, \alpha_3\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_3}{\alpha_7}}, q = \frac{\alpha_2}{\alpha_7}, r = 0, s = 0, t = -\frac{\alpha_1\sqrt[3]{\alpha_3}}{2\alpha_2\sqrt[3]{\alpha_7}} we have the family of representatives \langle \nabla_2+\nabla_3+\alpha\nabla_6+\nabla_7\rangle.

    4. \alpha_8\neq0 then by choosing r = -\frac{\alpha_7}{\alpha_8}q, s = \frac{\alpha_7^2-\alpha_5\alpha_8}{\alpha_8^2}q, t = -\frac{\alpha_2}{\alpha_8}x, we have \alpha_2^* = \alpha_5^* = \alpha_7^* = 0. Therefore, we can suppose that \alpha_2 = 0, \alpha_5 = 0, \alpha_7 = 0, then we have

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, then we have the representatives \langle \nabla_8 \rangle and \langle \nabla_6+\nabla_8 \rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_4}{\alpha_8}}, q = 1, r = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_4+\alpha\nabla_6+\nabla_8 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_3}{\alpha_8}}, q = 1, r = 0, s = 0, t = 0, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, \alpha_3 = 0, \alpha_4 = 0, then we have the representatives \langle \nabla_1+\nabla_8 \rangle and \langle \nabla_1+\nabla_6+\nabla_8 \rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(e)}}\ if \alpha_1\neq0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_4}{\alpha_8}}, q = \sqrt[6]{\frac{\alpha_1^3}{\alpha_4^2\alpha_8}}, r = 0, s = 0, t = 0, we have the family of representative \langle \nabla_1+\nabla_4+\alpha\nabla_6+\nabla_8 \rangle;

    {\rm{(f)}}\ if \alpha_1\neq0, \alpha_3\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_3}{\alpha_8}}, q = \frac{\sqrt{\alpha_1}}{\sqrt[4]{\alpha_3\alpha_8}}, r = 0, s = 0, t = 0, we have the family of representative \langle \nabla_1+\nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+ \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle^{O(\alpha,\beta) = O(i\alpha,-\beta) = O(-i\alpha,-\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_1+\nabla_3+\alpha\nabla_6 +\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_1+ \nabla_4+\nabla_5-\nabla_6 \rangle, \\ \langle \nabla_1+ \nabla_4+\alpha\nabla_6+\nabla_8 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \langle \nabla_1+\nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_1+\nabla_6 +\nabla_7 \rangle, \\ \langle \nabla_1+\nabla_6+\nabla_8 \rangle, \langle \nabla_1+\nabla_7 \rangle, \langle \nabla_1+\nabla_8 \rangle, \langle \nabla_2 \rangle, \langle \nabla_2+\nabla_3 \rangle, \langle \nabla_2+ \nabla_3+\nabla_6 \rangle, \\ \langle \nabla_2+\nabla_3+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_2+\nabla_4 \rangle, \langle \nabla_2+\nabla_4+\nabla_5-\nabla_6 \rangle, \\ \langle \nabla_2+\nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_2+\nabla_6 \rangle, \langle \nabla_2+ \nabla_6+\nabla_7 \rangle, \langle \nabla_2+\nabla_7 \rangle, \\ \langle \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle^{O(\alpha,\beta) = O(i\alpha,-\beta) = O(-i\alpha,-\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_3+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \langle \nabla_4+\nabla_5-\nabla_6 \rangle, \\ \langle \nabla_4+\alpha\nabla_6+\nabla_8 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \langle \nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_6+\nabla_7 \rangle, \langle \nabla_6+\nabla_8 \rangle, \\ \langle \nabla_7 \rangle, \langle \nabla_8 \rangle, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{188}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{189}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{190} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\&& e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{192} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{193} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{194} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{195} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{196} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ {\mathbf{N}}_{197} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ {\mathbf{N}}_{198} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{199}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{200} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{201} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{202}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{203} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{204} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{205} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{206}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\&& e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{207}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{208} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{209}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{210}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{211} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{212} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{213} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{214} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{15}^{4*}:

    \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{15} & \begin{array}{l} e_1e_2 = e_3 \\ e_1e_3 = e_4 \\ e_2e_2 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{15})& = &\\ {\langle [\Delta_{11}],[\Delta_{22}],[\Delta_{23}],[\Delta_{33}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{15}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{15})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle } \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&r&x^3&0\\ t&s&xr&x^4 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{8}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{15}) . Since

    \phi^T\begin{pmatrix} \alpha_1&0&0&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ 0&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{**}&\alpha_2^*\\ \alpha^{*}&\alpha_3^*+\alpha^{**}&\alpha^*_4&\alpha^*_5\\ \alpha^{**}&\alpha^*_4&\alpha^*_6&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix},

    we have

    \begin{array}{lcl} \alpha_1^* & = & \alpha_1x^2+2\alpha_2xt+\alpha_8t^2, \\ \alpha_2^* & = & (\alpha_2x+\alpha_8t)x^4, \\ \alpha_3^* & = & x^4 \alpha_3+2 r x^2 \alpha_4+2 s x^2 \alpha_5+r^2 \alpha_6+2 r s \alpha_7+ \\ &&{ s^2 \alpha_8-x (r x \alpha_2+t x^2 \alpha_7+r t \alpha_8), }\\ \alpha_4^* & = & (\alpha_4x^2+\alpha_6r+\alpha_7s)x^3+(\alpha_5x^2+\alpha_7r+\alpha_8s)xr, \\ \alpha_5^* & = & (\alpha_5x^2+\alpha_7r+\alpha_8s)x^4, \\ \alpha_6^* & = & (\alpha_6x^4+2\alpha_7x^2r+\alpha_8r^2)x^2, \\ \alpha_7^* & = & (\alpha_7x^2+\alpha_8r)x^5, \\ \alpha_8^* & = & \alpha_8x^8. \end{array}

    We are interested in (\alpha_2,\alpha_5,\alpha_7,\alpha_8)\neq(0,0,0,0) . Let us consider the following cases:

    1.\ \alpha_8 = 0, \alpha_7 = 0, \alpha_5 = 0, then \alpha_2\neq0 and we have

    {\rm{(a)}}\ if \alpha_6 = 0, \alpha_4 = 0, then by choosing x = 2\alpha_2, r = 4\alpha_2\alpha_3, s = 0, t = -\alpha_1, we have the representative \langle \nabla_2 \rangle;

    {\rm{(b)}}\ if \alpha_6 = 0, \alpha_4\neq0, \alpha_2 = 2\alpha_4, then we have the representatives \langle 2\nabla_2+\nabla_4 \rangle and \langle 2\nabla_2+\nabla_3+\nabla_4 \rangle depending on whether \alpha_3 = 0 or not;

    {\rm{(c)}}\ if \alpha_6 = 0, \alpha_4\neq0, \alpha_2\neq2\alpha_4, then by choosing x = \alpha_2-2\alpha_4, r = \alpha_3(\alpha_2-2\alpha_4), s = 0, t = \frac{\alpha_1(2\alpha_4-\alpha_2)}{2\alpha_2}, we have the family of representatives \langle \nabla_2+\alpha\nabla_4 \rangle_{\alpha\neq0,\frac{1}{2}}, which will be jointed with representatives from the cases (1a) and (1b);

    {\rm{(d)}}\ if \alpha_6\neq0, then by choosing x = \frac{\alpha_2}{\alpha_6}, r = -\frac{\alpha_2^2\alpha_4}{\alpha_6^3}, s = 0, t = -\frac{\alpha_1}{2\alpha_6}, we have the representative \langle \nabla_2+\alpha\nabla_3+\nabla_6 \rangle.

    2. \alpha_8 = 0, \alpha_7 = 0, \alpha_5\neq0 then we have

    {\rm{(a)}}\ if \alpha_5\neq-\alpha_6, then we have the following subcases:

    {\rm{(i)}}\ if \alpha_2 = 0, \alpha_1 = 0, then by choosing

    \begin{array}{c} x = 2 \alpha_5 (\alpha_5+\alpha_6), s = 2 \alpha_5 (\alpha_4^2 (2 \alpha_5+\alpha_6)-\alpha_3 (\alpha_5+\alpha_6)^2),\\ r = -4 \alpha_4 \alpha_5^2 (\alpha_5+\alpha_6),\end{array}

    we have the family of representatives \langle \nabla_5+\alpha\nabla_6 \rangle_{\alpha\neq-1} ;

    {\rm{(ii)}}\ if \alpha_2 = 0, \alpha_1\neq0, then by choosing

    x = \sqrt[4]{\frac{\alpha_1}{\alpha_5}}, r = -\frac{\alpha_4\sqrt{\alpha_1}}{(\alpha_5+\alpha_6)\sqrt{\alpha_5}}, s = \frac{((\alpha_5+\alpha_6)(2\alpha_4^2-\alpha_2\alpha_4)-\alpha_3(\alpha_5+\alpha_6)^2-\alpha_4^2\alpha_6)\sqrt{\alpha_1}}{2\alpha_5(\alpha_5+\alpha_6)^2\sqrt{\alpha_5}}, t = 0,

    we have the family of representatives \langle \nabla_1+\nabla_5+\alpha\nabla_6 \rangle_{\alpha\neq-1} ;

    {\rm{(iii)}}\ if \alpha_2\neq0, then by choosing

    \begin{array}{c} x = \frac{\alpha_2}{\alpha_5}, r = -\frac{\alpha_2^2\alpha_4}{\alpha_5^2(\alpha_5+\alpha_6)}, \\s = \frac{\alpha_2^2((\alpha_5+\alpha_6)(2\alpha_4^2-\alpha_2\alpha_4)-\alpha_3(\alpha_5+\alpha_6)^2-\alpha_4^2\alpha_6)}{2\alpha^3_5(\alpha_5+\alpha_6)^2}, t = -\frac{\alpha_1}{2\alpha_5}, \end{array}

    we have the family of representatives \langle \nabla_2+\nabla_5+\alpha\nabla_6 \rangle_{\alpha\neq-1} .

    (b) if \alpha_6 = -\alpha_5, then we have the following subcases:

    {\rm{(i)}}\ if \alpha_4 = 0, \alpha_2 = 0, \alpha_1 = 0, then we have the representative \langle \nabla_5-\nabla_6 \rangle, which will be jointed with the family from the case (2(a)i);

    {\rm{(ii)}}\ if \alpha_4 = 0, \alpha_2 = 0, \alpha_1\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_1}{\alpha_5}}, r = 0, s = -\frac{\alpha_3\sqrt{\alpha_1}}{2\alpha_5\sqrt{\alpha_4}}, t = 0, we have the representative \langle \nabla_1+\nabla_5-\nabla_6 \rangle, which will be jointed with the family from the case (2(a)ii);

    {\rm{(iii)}}\ if \alpha_4 = 0, \alpha_2\neq0, then by choosing x = \frac{\alpha_2}{\alpha_5}, r = 0, s = -\frac{\alpha^2_2\alpha_3}{2\alpha^3_5}, t = -\frac{\alpha_1}{2\alpha_5}, we have the representative \langle \nabla_2+\nabla_5-\nabla_6 \rangle, which will be jointed with the family from the case (2(a)iii);

    {\rm{(iv)}}\ if \alpha_4\neq0, then by choosing x = \frac{\alpha_4}{\alpha_5}, s = -\frac{\alpha_3 \alpha_4^2}{ 2 \alpha_5^3}, r = 0, we have the families of representatives

    \langle \alpha\nabla_1+\nabla_4+\nabla_5-\nabla_6 \rangle {\text{ and }} \langle \alpha\nabla_2+\nabla_4+\nabla_5-\nabla_6 \rangle_{\alpha\neq0}

    depending on \alpha_2 = 0 or not.

    3. \alpha_8 = 0, \alpha_7\neq0, then by choosing

    r = -\frac{\alpha_5}{\alpha_7}x^2, s = \frac{\alpha_5\alpha_6-\alpha_4\alpha_7}{\alpha_7^2}x^2, t = \frac{\alpha_3\alpha_7^2-2\alpha_4\alpha_5\alpha_7+\alpha_5^2\alpha_6+\alpha_2\alpha_5\alpha_7}{\alpha_7^3}x,

    we have \alpha_3^* = \alpha_4^* = \alpha_5^* = 0. Therefore, we can suppose that \alpha_3 = 0, \alpha_4 = 0, \alpha_5 = 0, and we have

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_2 = 0, then we have the representatives \langle \nabla_7 \rangle and \langle \nabla_6+\nabla_7 \rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_2\neq0, then by choosing x = \sqrt{\alpha_2\alpha_7^{-1}}, r = 0, s = 0, t = 0, we have the family of representative \langle \nabla_2+\alpha\nabla_6+\nabla_7 \rangle;

    {\rm{(c)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[5]{\frac{\alpha_1}{\alpha_7}}, r = 0, s = 0, t = 0, we have the family of representative \langle \nabla_1+\alpha\nabla_2+\beta\nabla_6+\nabla_7 \rangle.

    4. \alpha_8\neq0, then by choosing r = -\frac{\alpha_7}{\alpha_8}x^2, t = -\frac{\alpha_2}{\alpha_8}x, s = -\frac{\alpha_5x^2+\alpha_7r}{\alpha_8} we have \alpha_2^* = \alpha_5^* = \alpha_7^* = 0. Therefore, we can suppose that \alpha_2 = 0, \alpha_5 = 0, \alpha_7 = 0, then we have

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, then we have the representatives \langle \nabla_8 \rangle and \langle \nabla_6+\nabla_8 \rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[3]{{\alpha_4}{\alpha_8^{-1}}}, r = 0, s = 0, t = 0, we have the family of representative \langle \nabla_4+\alpha\nabla_6+\nabla_8 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \sqrt[4]{{\alpha_3}{\alpha_8^{-1}}}, r = 0, s = 0, t = 0, we have the family of representative \langle \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[6]{{\alpha_1}{\alpha_8^{-1}}}, r = 0, s = 0, t = 0, we have the family of representative \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\nabla_8 \rangle.

    Summarizing all cases we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\beta\nabla_6+\nabla_7 \rangle ^{O(\alpha,\beta) = O(\eta_5^2\alpha,-\eta_5\beta) = O(\eta_5^4\alpha,\eta_5^2\beta) = O(-\eta_5\alpha,-\eta_5^3\beta) = O(-\eta_5^3\alpha,\eta_5^4\beta)}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4 +\gamma\nabla_6+\nabla_8 \rangle^{ { \begin{array}{l}O(\alpha,\beta,\gamma) = O(-\eta_3\alpha,\beta,\eta_3^2\gamma) = O(-\eta_3\alpha,-\beta,\eta_3^2\gamma) = \\ O(\eta_3^2\alpha,-\beta,-\eta_3\gamma) = O(\eta_3^2\alpha,\beta,-\eta_3\gamma) = O(\alpha,-\beta,\gamma) \end{array}}}, \\ \langle \alpha\nabla_1+ \nabla_4+\nabla_5-\nabla_6 \rangle, \langle \nabla_1+\nabla_5+\alpha\nabla_6 \rangle, \langle 2\nabla_2+\nabla_3 +\nabla_4 \rangle, \langle \nabla_2+\alpha\nabla_3+\nabla_6 \rangle, \\ \langle \nabla_2+\alpha\nabla_4 \rangle, \langle \alpha\nabla_2+\nabla_4+\nabla_5-\nabla_6 \rangle_{\alpha\neq0}, \langle \nabla_2+\nabla_5+\alpha\nabla_6 \rangle, \\ \langle \nabla_2+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_3+\alpha\nabla_4+\beta\nabla_6+\nabla_8 \rangle ^{O(\alpha,\beta) = O(i\alpha,-\beta) = O(-i\alpha,-\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_4+\alpha\nabla_6+\nabla_8 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_5+\alpha\nabla_6 \rangle, \langle \nabla_6+ \nabla_7 \rangle, \langle \nabla_6+\nabla_8 \rangle, \\ \langle \nabla_7 \rangle, \langle \nabla_8 \rangle, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{215}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{216}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4+\alpha e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{217}^{\alpha} & : & e_1e_1 = \alpha e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{218}^\alpha & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\& & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{219} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = 2e_5 & e_2e_2 = e_4+e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{220}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_4+\alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{221}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{222}^{\alpha\neq 0} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{223}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{224}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\&& e_2e_2 = e_4 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{225}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & { e_2e_2 = e_4+e_5 } \\&& e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{226}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\&& e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{227}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{228} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{229} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{230} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{231} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{16}^{4*}:

    {\mathbf{N}}^{4*}_{19} \begin{array}{ll} e_1e_1 = e_4 & e_1e_2 = e_3 \\ e_2e_2 = e_4 & e_3e_3 = e_4 \end{array} \begin{array}{l} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{19}) = \langle [\Delta_{11}],[\Delta_{13}],[\Delta_{22}],[\Delta_{23}]\rangle\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{19}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{19})\oplus \langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle \end{array}
    { \begin{array}{l} \begin{array}{l} \phi_1 = \begin{pmatrix} x&0&0&0\\ 0 & q&0&0\\ 0&0 & xq&0\\ t & s&0&1 \end{pmatrix}, \\ {x^2 = 1,q^2 = 1} \end{array} \begin{array}{l} \phi_2 = \begin{pmatrix} 0 & p&0&0\\ y&0&0&0\\ 0&0 & yp&0\\ t & s&0&1 \end{pmatrix}, \\ {y^2 = 1, p^2 = 1} \end{array} \end{array} }

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    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{33}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{8}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{16}) . Since

    \phi^T\begin{pmatrix} \alpha_1&0&0&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ 0&\alpha_4&\alpha_6&\alpha_7\\ \alpha_2&\alpha_5&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{**}&\alpha_2^*\\ \alpha^{*}&\alpha^*_3&\alpha^*_4+\alpha^{**}&\alpha_5^*\\ \alpha^{**}&\alpha^*_4+\alpha^{**}&\alpha_6^*&\alpha^*_7\\ \alpha^*_2&\alpha^*_5&\alpha^*_7&\alpha^*_8 \end{pmatrix},

    in the case \phi = \phi_1 , we have

    \begin{array}{ll} \alpha_1^* = \alpha_1x^2+2\alpha_2xt+\alpha_8t^2, & \alpha_2^* = (\alpha_2x+\alpha_8t)x^3, \\ \alpha_3^* = \alpha_3x^2+2\alpha_5xs+\alpha_8s^2, & \alpha_4^* = (\alpha_4x+\alpha_7s)x^2-\alpha_7x^2t, \\ \alpha_5^* = (\alpha_5x+\alpha_8s)x^3, & \alpha_6^* = \alpha_6x^4, \\ \alpha_7^* = \alpha_7x^5, & \alpha_8^* = \alpha_8x^6; \end{array}

    and on the opposite case, for \phi = \phi_2, we have

    \begin{array}{ll} \alpha_1^* = \alpha_3y^2+2 \alpha_5 t y+ \alpha_8 t^2, & \alpha_2^* = (\alpha_5 y + \alpha_8t )y^3, \\ \alpha_3^* = \alpha_1y^2+2 \alpha_2s y + \alpha_8s^2, & \alpha_4^* = ((s-t) \alpha_7-y \alpha_4)y^2, \\ \alpha_5^* = (y \alpha_2+s \alpha_8)y^3, & \alpha_6^* = \alpha_6 y^4, \\ \alpha_7^* = \alpha_7y^5, & \alpha_8^* = \alpha_8 y^6. \end{array}

    We are interested in (\alpha_2,\alpha_5,\alpha_7,\alpha_8)\neq(0,0,0,0) . Let us consider the following cases:

    1.\ \alpha_8 = 0, \alpha_7 = 0, \alpha_5 = 0, then \alpha_2\neq0 and

    {\rm{(a)}}\ if \alpha_4\neq 0, then by choosing \phi = \phi_1, x = \alpha_4 \alpha_2^{-1}, t = -\frac{ \alpha_1 \alpha_4 }{2 \alpha_2^{2}}, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+\nabla_4 +\beta \nabla_6 \rangle;

    {\rm{(b)}}\ if \alpha_4 = 0,\alpha_3\neq0, then by choosing \phi = \phi_1, x = \sqrt{ \alpha_3 \alpha_2^{-1}}, t = -\frac{ \alpha_1 \sqrt{ \alpha_3}}{2 \sqrt{\alpha_2^3}}, we have the family of representatives \langle \nabla_2+\nabla_3+\alpha \nabla_6 \rangle;

    {\rm{(c)}}\ if \alpha_4 = 0,\alpha_3 = 0, then by choosing \phi = \phi_1, x = 2 \alpha_2, t = - \alpha_1, s = 0, we have the family of representatives \langle \nabla_2+ \alpha \nabla_6 \rangle.

    2. \alpha_8 = 0, \alpha_7 = 0, \alpha_5\neq0 and

    {\rm{(a)}}\ if \alpha_2\neq 0,\alpha_4\neq 0, then by choosing

    \phi = \phi_1, x = \frac{\alpha_4}{\alpha_5}, t = -\frac{\alpha_1 \alpha_4}{ 2 \alpha_2 \alpha_5}, s = -\frac{ \alpha_3 \alpha_4}{ 2 \alpha_5^2},

    we have the following family of representatives

    \langle \alpha \nabla_2+\nabla_4+\nabla_5+ \beta \nabla_6 \rangle_{\alpha\neq0};

    {\rm{(b)}}\ if \alpha_2\neq 0,\alpha_4 = 0, then by choosing

    \phi = \phi_1, x = 2 \alpha_2 \alpha_5, t = - \alpha_1 \alpha_5, s = - \alpha_2 \alpha_3,

    we have the following family of representatives \langle \alpha \nabla_2+ \nabla_5+ \beta \nabla_6 \rangle_{\alpha\neq0};

    {\rm{(c)}}\ if \alpha_2 = 0, then by choosing \phi = \phi_2, y = 1, t = 0, s = 0, we have the representative with \alpha_5^* = 0 and \alpha_2^*\neq0, which was considered above.

    3. \alpha_8 = 0, \alpha_7\neq0, then we have

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_5 = 0, \alpha_1 = 0, \alpha_3 = 0, then we have the representatives \langle \nabla_7 \rangle and \langle \nabla_6+\nabla_7\rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_5 = 0, \alpha_1\neq0, then by choosing

    \phi = \phi_1, x = \sqrt[3]{{\alpha_1}{\alpha_7^{-1}}}, s = 0, t = {\alpha_4\sqrt[3]{\alpha_1} \alpha_7^{-1}},

    we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_6+\nabla_7\rangle;

    {\rm{(c)}}\ if \alpha_2\neq0, then by choosing

    \phi = \phi_1, x = {\alpha_2}{\alpha_7^{-1}}, s = -({\alpha_1\alpha_7+2\alpha_2\alpha_4}) /(2\alpha_7^2), t = -{\alpha_1}/ ({2\alpha_7}),

    we have the family of representatives \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5+\gamma\nabla_6+\nabla_7\rangle.

    4. \alpha_8\neq0, then by choosing \phi = \phi_1, t = -\frac{\alpha_2}{\alpha_8}x, s = -\frac{\alpha_5}{\alpha_8}x, we have \alpha_2^* = \alpha_5^* = 0. Now we can suppose that \alpha_2 = 0, \alpha_5 = 0 and we have

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, \alpha_6 = 0, then we have the representatives \langle \nabla_8\rangle and \langle \nabla_7+\nabla_8\rangle depending on whether \alpha_7 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, \alpha_6\neq0, then by choosing \phi = \phi_1, x = \sqrt{{\alpha_6}{\alpha_8^{-1}}}, s = 0, t = 0, we have the family of representative \langle \nabla_6+\alpha\nabla_7+\nabla_8\rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing \phi = \phi_1, x = \sqrt[3]{{\alpha_4}{\alpha_8^{-1}}}, s = 0, t = 0, we have the family of representatives \langle \nabla_4+\alpha\nabla_6+\beta\nabla_7+\nabla_8\rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, then by choosing \phi = \phi_1, x = \sqrt[4]{{\alpha_1}{\alpha_8^{-1}}}, s = 0, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\mu\nabla_7+\nabla_8\rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\mu\nabla_7+\nabla_8\rangle ^{ { \begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(\alpha,i\beta,-\gamma,-i\mu) = \\ O(\alpha,-i\beta,-\gamma,i\mu) = O(\alpha,-\beta,\gamma,-\mu) = \\ O(\frac{1}{\alpha},-\frac{\beta}{\sqrt[4]{\alpha^{3}}}, \frac{\gamma}{\sqrt{\alpha}},\frac{\mu}{\sqrt[4]{\alpha}}) = \\ O(\frac{1}{\alpha},-\frac{i\beta}{ \sqrt[4]{\alpha^{3}}},-\frac{\gamma}{\sqrt{\alpha}},-\frac{i\mu}{\sqrt[4]{\alpha}}) = \\ O(\frac{1}{\alpha},\frac{\beta}{\sqrt[4]{\alpha^{3}}}, \frac{\gamma}{\sqrt{\alpha}},-\frac{\mu}{\sqrt[4]{\alpha}}) = O(\frac{1}{\alpha},-\frac{\beta}{\sqrt[4]{\alpha^{3}}}, \frac{\gamma}{\sqrt{\alpha}},\frac{\mu}{\sqrt[4]{\alpha}})\end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_6+\nabla_7\rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(\alpha,-\eta_3\beta) = O(\alpha,\eta_3^2\beta) = \\ O(\alpha^{-1},-\eta_3\beta\sqrt[3]{\alpha^{-1}}) = O(\alpha^{-1},\eta_3^2\beta\sqrt[3]{\alpha^{-1}}) = O(\alpha^{-1},\beta\sqrt[3]{\alpha^{-1}})\end{array}}}, \\ \langle \nabla_2+\alpha\nabla_3+\nabla_4 +\beta \nabla_6 \rangle, \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5+\gamma\nabla_6+\nabla_7\rangle^{O(\alpha,\beta,\gamma) = O(-\frac{\alpha}{\beta^4},\frac{1}{\beta},\frac{\gamma}{\beta})}, \\ \langle \nabla_2+\nabla_3+\alpha \nabla_6 \rangle, \langle \alpha \nabla_2+\nabla_4+\nabla_5+ \beta \nabla_6 \rangle_{\alpha\neq0}^{O(\alpha,\beta) = O(\alpha^{-1},\beta\alpha^{-1})}, \\ \langle \alpha \nabla_2+ \nabla_5+ \beta \nabla_6 \rangle_{\alpha\neq0}^{O(\alpha,\beta) = O(\alpha^{-1},\beta\alpha^{-1})}, \langle \nabla_2+ \alpha \nabla_6 \rangle, \\ \langle \nabla_4+\alpha\nabla_6+\beta\nabla_7+\nabla_8\rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(\eta_3^2\alpha,-\eta_3\beta) = O(-\eta_3\alpha,\eta_3^2\beta) = \\ O(\eta_3^2\alpha,\eta_3\beta) = O(-\eta_3\alpha,-\eta_3^2\beta) = O(\alpha,-\beta)\end{array} }}, \langle \nabla_6+\nabla_7\rangle, \\ \langle \nabla_6+\alpha\nabla_7+\nabla_8\rangle^{O(\alpha) = O(-\alpha)}, \langle \nabla_7 \rangle, \langle \nabla_7+\nabla_8\rangle, \langle \nabla_8\rangle, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{232}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{233}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{234}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4+e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{235}^{\alpha, \beta, \gamma} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{236}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{237}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_3 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{238}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{239}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{240}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4+e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{241} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{242}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{243} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{244} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{245} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{17}^{4*}:

    {\mathbf{N}}^{4*}_{19} \begin{array}{ll} e_1e_1 = e_4 & e_1e_2 = e_3 \\ e_2e_2 = e_4 & e_3e_3 = e_4 \end{array} \begin{array}{l} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{19}) = \langle [\Delta_{11}],[\Delta_{13}],[\Delta_{22}],[\Delta_{23}]\rangle\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{19}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{19})\oplus \langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle \end{array}
    { \begin{array}{l} \begin{array}{l} \phi_1 = \begin{pmatrix} x&0&0&0\\ 0 & q&0&0\\ 0&0 & xq&0\\ t & s&0&1 \end{pmatrix}, \\ {x^2 = 1,q^2 = 1} \end{array} \begin{array}{l} \phi_2 = \begin{pmatrix} 0 & p&0&0\\ y&0&0&0\\ 0&0 & yp&0\\ t & s&0&1 \end{pmatrix}, \\ {y^2 = 1, p^2 = 1} \end{array} \end{array} }

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    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{22}], \\ \nabla_5 = [\Delta_{23}], & \nabla_6 = [\Delta_{24}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{8}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{17}) . Since

    \phi^T\begin{pmatrix} \alpha_1&0&\alpha_2&\alpha_3\\ 0&\alpha_4&\alpha_5&\alpha_6\\ \alpha_2&\alpha_5&0&\alpha_7\\ \alpha_3&\alpha_6&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha^{*}&\alpha^*_4&\alpha^*_5&\alpha_6^*\\ \alpha^{*}_2&\alpha^*_5& 0 &\alpha^*_7\\ \alpha^*_3&\alpha^*_6&\alpha^*_7&\alpha^*_8 \end{pmatrix},

    then in the case \phi = \phi_1, we have

    \begin{array}{ll} \alpha_1^* = \alpha_1x^2+2\alpha_3xt+\alpha_8t^2, & \alpha_2^* = (\alpha_2x+\alpha_7t)xq, \\ \alpha_3^* = (\alpha_3x+\alpha_8t)x^2q^2, & \alpha_4^* = \alpha_4q^2+2\alpha_6qs+\alpha_8s^2, \\ \alpha_5^* = (\alpha_5q+\alpha_7s)xq, & \alpha_6^* = (\alpha_6q+\alpha_8s)x^2q^2, \\ \alpha_7^* = \alpha_7x^3q^3, & \alpha_8^* = \alpha_8x^4q^4; \end{array}

    and in the opposite case \phi = \phi_2, we have

    \begin{array}{ll} \alpha_1^* = \alpha_4p^2+2 \alpha_6p t+ \alpha_8t^2, & \alpha_2^* = ( \alpha_5p + \alpha_7t)p y, \\ \alpha_3^* = ( \alpha_6p+ \alpha_8t)p^2 y^2 , & \alpha_4^* = \alpha_1y^2+2 \alpha_3s y+ \alpha_8s^2, \\ \alpha_5^* = ( \alpha_2y+ \alpha_7s)p y, & \alpha_6^* = ( \alpha_3y+ \alpha_8s)p^2 y^2 , \\ \alpha_7^* = \alpha_7p^3 y^3, & \alpha_8^* = \alpha_8p^4 y^4. \end{array}

    We are interested in (\alpha_3,\alpha_6,\alpha_7,\alpha_8)\neq(0,0,0,0) . Let us consider the following cases:

    1.\ \alpha_8 = 0, \alpha_7 = 0, \alpha_6 = 0, then \alpha_3\neq0 and choosing \phi = \phi_1, t = -\frac{\alpha_1}{2\alpha_3}x, we get \alpha_1^* = 0. Now consider the following subcases:

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_4 = 0, \alpha_5 = 0, then we have the representative \langle \nabla_3 \rangle;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_4 = 0, \alpha_5\neq0, then by choosing \phi = \phi_1, x = \sqrt{\frac{\alpha_5}{\alpha_3}}, q = 1, s = 0, t = -\frac{\alpha_1\sqrt{\alpha_5}}{2\alpha_3\sqrt{\alpha_3}}, we have the representative \langle \nabla_3+\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_2 = 0, \alpha_4\neq0, then by choosing \phi = \phi_1, x = \sqrt[3]{\frac{\alpha_4}{\alpha_3}}, q = 1, s = 0, t = -\frac{\alpha_1\sqrt[3]{\alpha_5}}{2\alpha_3\sqrt[3]{\alpha_3}}, we have the representative \langle \nabla_3+\nabla_4+\alpha\nabla_5 \rangle;

    {\rm{(d)}}\ if \alpha_2\neq0, \alpha_4 = 0, \alpha_5 = 0, then by choosing \phi = \phi_1, x = \alpha_2, q = \frac{1}{\alpha_3}, s = 0, t = -\frac{\alpha_1\alpha_2}{2\alpha_3}, we have the representative \langle \nabla_2+\nabla_3 \rangle;

    {\rm{(e)}}\ if \alpha_2\neq0, \alpha_4 = 0, \alpha_5\neq0, then by choosing \phi = \phi_1, x = \sqrt{\frac{\alpha_5}{\alpha_3}}, q = \sqrt{\frac{\alpha_2^2}{\alpha_3\alpha_5}}, s = 0, t = -\frac{\alpha_1\sqrt{\alpha_5}}{2\alpha_3\sqrt{\alpha_3}}, we have the representative \langle \nabla_2+\nabla_3+\nabla_5 \rangle;

    {\rm{(f)}}\ if \alpha_2\neq0, \alpha_4\neq0, then by choosing \phi = \phi_1, x = \sqrt[3]{{\alpha_4}{\alpha_3^{-3}}}, q = \sqrt[3]{{\alpha_2^3}{\alpha^{-2}_3\alpha_4^{-1}}}, we have the family of representative \langle \nabla_2+\nabla_3+\nabla_4+\alpha\nabla_5 \rangle.

    2. \alpha_8 = 0, \alpha_7 = 0, \alpha_6\neq0, and \alpha_3 = 0 , then by choosing some suitable automorphism \phi_2 we have \alpha_3^*\neq0 which is the case considered above. Now we can suppose that \alpha_3\neq0, and choosing t = -\frac{\alpha_1}{2\alpha_3}x, s = -\frac{\alpha_4}{2\alpha_6}x, we have \alpha_1^* = 0, \alpha_4^* = 0. Therefore, we can suppose that \alpha_1 = 0, \alpha_4 = 0. Consider the following subcases:

    {\rm{(a)}}\ \alpha_2 = 0, \alpha_5 = 0, then by choosing \phi = \phi_1, x = \alpha_6, q = \alpha_3, s = 0, t = 0, we have the representative \langle \nabla_3+\nabla_6 \rangle;

    {\rm{(b)}}\ \alpha_2\neq0, then by choosing \phi = \phi_1, x = {\alpha_3^{-1}}{\sqrt{\alpha_2\alpha_6}}, q = \sqrt{\alpha_2\alpha_6^{-1}}, s = 0, t = 0, we have the family of representatives \langle \nabla_2+\nabla_3+\alpha\nabla_5+\nabla_6 \rangle.

    3. \alpha_8 = 0, \alpha_7\neq0, then by choosing \phi = \phi_1, t = -{\alpha_2}{\alpha_7^{-1}}x, s = -{\alpha_5}{\alpha_7^{-1}}q, we have \alpha_2^* = 0, \alpha_5^* = 0. Therefore, we can suppose that \alpha_2 = 0, \alpha_5 = 0. Consider the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_3 = 0, \alpha_6 = 0, then we have the representative \langle \nabla_7 \rangle ;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_3 = 0, \alpha_6\neq0, then by choosing \phi = \phi_1, x = {\alpha_6}{\alpha_7^{-1}}, q = 1, s = 0, t = 0, we have the representative \langle \nabla_6+\nabla_7 \rangle ;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_3\neq0, and \alpha_6 = 0, then by choosing some suitable automorphism \phi_2, we have \alpha_6^*\neq0 . Thus we can consider the case \alpha_6\neq0 and choosing \phi = \phi_1, x = {\alpha_6}{\alpha_7^{-1}}, q = {\alpha_3}{\alpha_7^{-1}}, s = 0, t = 0, we have the representative \langle \nabla_3+\nabla_6+\nabla_7 \rangle ;

    {\rm{(d)}}\ if \alpha_1 = 0, \alpha_4\neq0, \alpha_3 = 0, \alpha_6 = 0, then by choosing \phi = \phi_1, x = 1, q = {\alpha_4}{\alpha_7^{-1}}, s = 0, t = 0, we have the representative \langle \nabla_4+\nabla_7 \rangle;

    {\rm{(e)}}\ if \alpha_1 = 0, \alpha_4\neq0, \alpha_3 = 0, \alpha_6\neq0, then by choosing \phi = \phi_1, x = {\alpha_6}{\alpha_7}^{-1}, q = {\alpha_4\alpha_7^2}{\alpha_6^{-3}}, we have the representative \langle \nabla_4+\nabla_6+\nabla_7 \rangle;

    {\rm{(f)}}\ if \alpha_1 = 0, \alpha_4\neq0, \alpha_3\neq0, then by choosing

    \phi = \phi_1, x = \sqrt[3]{{\alpha_4}{\alpha_3^{-1}}}, q = {\alpha_3}{\alpha_7^{-1}}, s = 0, t = 0,

    we have the family of representatives \langle \nabla_3+\nabla_4+\alpha\nabla_6+\nabla_7 \rangle ;

    {\rm{(g)}}\ if \alpha_1\neq0. In case of \alpha_4 = 0, choosing some suitable automorphism \phi_2, we have \alpha_4^*\neq0. Thus, we can suppose \alpha_4\neq0 , and choosing

    \phi = \phi_1, x = \sqrt[8]{{\alpha^3_4}{\alpha_1^{-1}\alpha_7^{-2}}}, q = \sqrt[8]{{\alpha_1^3}{\alpha_4^{-1}\alpha^{-2}_7}}, s = 0, t = 0,

    we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\nabla_4+\beta\nabla_6+\nabla_7 \rangle .

    4. \alpha_8\neq0, then by choosing \phi = \phi_1, t = -\frac{\alpha_3}{\alpha_8}x, s = -\frac{\alpha_6}{\alpha_8}q, we have \alpha_3^* = 0, \alpha_6^* = 0. Consider the following cases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_2 = 0, \alpha_5 = 0, then we have the representatives \langle \nabla_8 \rangle and \langle \nabla_7+\nabla_8 \rangle depending on whether \alpha_7 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_2 = 0, \alpha_5\neq0, then we have the representatives \langle \nabla_5+\nabla_8 \rangle and \langle \nabla_5+\nabla_7+\nabla_8 \rangle depending on whether \alpha_7 = 0 or not;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_2\neq0. In case of \alpha_5 = 0, choosing some suitable automorphism \phi_2, we have \alpha_5^*\neq0 . Thus, we can suppose \alpha_5\neq0 , and choosing

    \phi = \phi_1, x = \sqrt[5]{{\alpha^3_5}{\alpha_2^{-2}\alpha_8^{-1}}}, q = \sqrt[5]{{\alpha^3_2}{\alpha_5^{-2}\alpha_8^{-1}}}, s = 0, t = 0,

    we have the family of representatives \langle \nabla_2+\nabla_5+\alpha\nabla_7+\nabla_8 \rangle ;

    {\rm{(d)}}\ if \alpha_1 = 0, \alpha_4\neq0, \alpha_2 = 0, \alpha_5 = 0, then we have the representatives \langle \nabla_4+\nabla_8 \rangle and \langle \nabla_4+\nabla_7+\nabla_8 \rangle depending on whether \alpha_7 = 0 or not;

    {\rm{(e)}}\ if \alpha_1 = 0, \alpha_4\neq0, \alpha_2 = 0, \alpha_5\neq0 , then by choosing

    \phi = \phi_1, x = {\alpha_4}{\alpha_5^{-1}}, q = {\alpha^2_5}{\alpha_4^{-1}\sqrt{\alpha_4^{-1}\alpha_8^{-1}}}, s = 0, t = 0,

    we have the family of representatives \langle \nabla_4+\nabla_5+\alpha\nabla_7+\nabla_8 \rangle ;

    {\rm{(f)}}\ if \alpha_1 = 0, \alpha_4\neq0, \alpha_2\neq0 , then by choosing

    \phi = \phi_1, x = \sqrt[8]{{\alpha_4^3}{\alpha_2^{-2}\alpha_8^{-1}}}, q = \sqrt[4]{{\alpha^2_2}{\alpha_4^{-1}\alpha_8^{-1}}}, s = 0, t = 0,

    we have the family of representatives \langle \nabla_2+\nabla_4+\alpha\nabla_5+\beta\nabla_7+\nabla_8 \rangle ;

    {\rm{(g)}}\ if \alpha_1\neq0 then by choosing some suitable automorphism \phi_2, we have \alpha_4^*\neq0 . Thus, we can suppose \alpha_4\neq0 , and choosing

    \phi = \phi_1, x = \sqrt[6]{{\alpha^2_4}{\alpha_1^{-1}\alpha_8^{-1}}}, q = \sqrt[6]{{\alpha_1^2}{\alpha_4^{-1}\alpha_8^{-1}}}, s = 0, t = 0,

    we have the family of representatives

    \langle \nabla_1+\alpha\nabla_2+\nabla_4+\beta\nabla_5+\gamma\nabla_7+\nabla_8 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\nabla_4+\beta\nabla_5+\gamma\nabla_7+\nabla_8 \rangle^{ { \begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_3^2\alpha, \eta_3^2\beta, \eta_3^2\gamma) = O(-\eta_3^2\alpha, \eta_3^2\beta, -\eta_3^2\gamma) = \\ O(\eta_3^2\alpha, -\eta_3^2\beta, -\eta_3^2\gamma) = O(-\eta_3^2\alpha, \eta_3^2\beta, \eta_3^2\gamma) = \\ O(\eta_3\alpha, \eta_3\beta, -\eta_3\gamma) = O(-\eta_3\alpha, \eta_3\beta, \eta_3\gamma) = \\ O(\eta_3\alpha, -\eta_3\beta, \eta_3\gamma) = O(-\eta_3\alpha, -\eta_3\beta, -\eta_3\gamma) = \\ O(-\alpha, \beta, - \gamma) = O(\alpha, -\beta, - \gamma) = \\ O(-\alpha, -\beta, \gamma) = O(\beta,\alpha,\gamma) = \\ O(\eta_3^2\beta, \eta_3^2\alpha, \eta_3^2\gamma) = O(-\eta_3^2\beta, \eta_3^2\alpha, -\eta_3^2\gamma) = \\ O(\eta_3^2\beta, -\eta_3^2\alpha, -\eta_3^2\gamma) = O(-\eta_3^2\beta, \eta_3^2\alpha, \eta_3^2\gamma) = \\ O(\eta_3\beta, \eta_3\alpha, -\eta_3\gamma) = O(-\eta_3\beta, \eta_3\alpha, \eta_3\gamma) = \\ O(\eta_3\beta, -\eta_3\alpha, \eta_3\gamma) = O(-\eta_3\beta, -\eta_3\alpha, -\eta_3\gamma) = \\ O(-\beta, \alpha, - \gamma) = O(\beta, -\alpha, - \gamma) = O(-\beta, -\alpha, \gamma) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\nabla_4+\beta\nabla_6 +\nabla_7 \rangle^{ {\begin{array}{l} O(\alpha,\beta) = O(\eta_4\alpha,-\eta_4\beta) = O(-\eta_4\alpha,\eta_4\beta) = O(\eta_4^3\alpha,-\eta_4^3\beta) = \\ O(-\eta_4^3\alpha,\eta_4^3\beta) = O(-i\alpha,-i\beta) = O(i\alpha,i\beta) = O(-\alpha,-\beta) = \\ O(\beta,\alpha) = O(\eta_4\beta,-\eta_4\alpha) = O(-\eta_4\beta,\eta_4\alpha) = O(\eta_4^3\beta,-\eta_4^3\alpha) = \\ O(-\eta_4^3\beta,\eta_4^3\alpha) = O(-i\beta,-i\alpha) = O(i\beta,i\alpha) = O(-\beta,-\alpha) \end{array}}}, \\ \langle \nabla_2+\nabla_3 \rangle, \langle \nabla_2+\nabla_3+\nabla_4+\alpha\nabla_5 \rangle^{O(\alpha) = O(-\eta_3 \alpha) = O(\eta_3^2 \alpha)}, \\ \langle \nabla_2+\nabla_3+\nabla_5 \rangle, \langle \nabla_2+\nabla_3 +\alpha\nabla_5+\nabla_6 \rangle^{O(\alpha) = O(\alpha^{-1})}, \\ \langle \nabla_2+\nabla_4+\alpha\nabla_5+\beta\nabla_7+\nabla_8 \rangle ^{{ \begin{array}{l} O(\alpha,\beta) = O(\eta_4^3\alpha,-\eta_4^3\beta) = O(-\eta_4^3\alpha,\eta_4^3\beta) = O(\eta_4\alpha,-\eta_4\beta) = \\ O(-\eta_4\alpha,\eta_4\beta) = O(i\alpha,i\beta) = O(-i\alpha,-i\beta) = O(-\alpha,-\beta) \end{array}}}, \\ \langle \nabla_2+\nabla_5+\alpha\nabla_7+\nabla_8 \rangle ^{ { \begin{array}{l} O(\alpha) = O(\eta_5^2\alpha) = O(\eta_5^4\alpha) = \\ O(-\eta_5\alpha) = O(-\eta_5^3\alpha) \end{array}}}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+\nabla_4+\alpha\nabla_5 \rangle ^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \\ \langle \nabla_3+\nabla_4+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_3+\nabla_5 \rangle, \langle \nabla_3+\nabla_6 \rangle, \\ \langle \nabla_3+\nabla_6+\nabla_7 \rangle, \langle \nabla_4+\nabla_5+\alpha\nabla_7+\nabla_8 \rangle^{O(\alpha) = O(-\alpha)}, \langle \nabla_4+\nabla_6+\nabla_7 \rangle, \\ \langle \nabla_4+\nabla_7 \rangle, \langle \nabla_4+\nabla_7+\nabla_8 \rangle, \langle \nabla_4+\nabla_8 \rangle, \langle \nabla_5+\nabla_7+\nabla_8 \rangle, \langle \nabla_5+\nabla_8 \rangle, \\ \langle \nabla_6+\nabla_7 \rangle, \langle \nabla_7 \rangle, \langle \nabla_7+\nabla_8 \rangle, \langle \nabla_8 \rangle, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{246}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{247}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{248} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{249}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{250} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{251}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{252}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{253}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{254} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{255}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{256}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{257} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{258} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{259} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{260}^{\alpha} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{261} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{262} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{263} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{264} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{265} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{266} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{267} & : & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{268} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{269} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{270} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{18}^{4*}:

    \begin{array}{|l|l|l|l|} \hline {\mathbf{N}}^{4*}_{18} & \begin{array}{l} e_1e_1 = e_4 \\ e_1e_2 = e_3 \\ e_3e_3 = e_4 \end{array} & \begin{array}{lcl} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{18})& = &\\ {\langle [\Delta_{11}],[\Delta_{13}],[\Delta_{22}],[\Delta_{23}]\rangle}\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{18})& = &\mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{18})\oplus\\ {\langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle} \end{array} & \phi_{\pm} = \begin{pmatrix} x&0&0&0\\ 0&\pm 1&0&0\\ 0&0&\pm x &0\\ t&s&0&x^2 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{22}], \\ \nabla_5 = [\Delta_{23}], & \nabla_6 = [\Delta_{24}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{8}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{18}) . Since

    \phi_{\pm}^T\begin{pmatrix} \alpha_1&0&\alpha_2&\alpha_3\\ 0&\alpha_4&\alpha_5&\alpha_6\\ \alpha_2&\alpha_5&0&\alpha_7\\ \alpha_3&\alpha_6&\alpha_7&\alpha_8 \end{pmatrix}\phi_{\pm} = \begin{pmatrix} \alpha_1^*&\alpha{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha{*}&\alpha^*_4&\alpha^*_5&\alpha_6^*\\ \alpha^{*}_2&\alpha^*_5&0&\alpha^*_7\\ \alpha^*_3&\alpha^*_6&\alpha^*_7&\alpha^*_8 \end{pmatrix},

    we have

    \begin{array}{ll} \alpha_1^* = \alpha_1x^2+2\alpha_3xt+\alpha_8t^2, & \alpha_2^* = \pm (\alpha_2x+\alpha_7t) x, \\ \alpha_3^* = (\alpha_3x+\alpha_8t)x^2, & \alpha_4^* = \alpha_4\pm2\alpha_6s+\alpha_8s^2, \\ \alpha_5^* = (\alpha_5\pm\alpha_7s)x, & \alpha_6^* = (\pm\alpha_6+\alpha_8s)x^2, \\ \alpha_7^* = \pm\alpha_7x^3, & \alpha_8^* = \alpha_8x^4. \end{array}

    We are interested in (\alpha_3,\alpha_6,\alpha_7,\alpha_8)\neq(0,0,0,0) . Let us consider \phi = \phi_+ and the following cases:

    1.\ \alpha_8 = 0, \alpha_7 = 0, \alpha_6 = 0, then \alpha_3\neq0 and choosing t = -\frac{\alpha_1}{2\alpha_3}x, we get \alpha_1^* = 0. Now consider the following subcases:

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_4 = 0, \alpha_5 = 0, then we have the representative \langle \nabla_3 \rangle ;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_4 = 0, \alpha_5\neq0, then by choosing x = \sqrt{\frac{\alpha_5}{\alpha_3}}, s = 0, t = -\frac{\alpha_1\sqrt{\alpha_5}}{2\alpha_3\sqrt{\alpha_3}}, we have the representative \langle \nabla_3+\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_2 = 0, \alpha_4\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_4}{\alpha_3}}, s = 0, t = -\frac{\alpha_1\sqrt[3]{\alpha_4}}{2\alpha_3\sqrt[3]{\alpha_3}}, we have the family of representatives \langle \nabla_3+\nabla_4+\alpha\nabla_5 \rangle;

    {\rm{(d)}}\ if \alpha_2\neq0, then by choosing x = {\alpha_2}{\alpha_3^{-1}}, s = 0, t = -\frac{\alpha_1\alpha_2}{2\alpha^2_3}, we have the family of representatives \langle \nabla_2+\nabla_3+\alpha\nabla_4+\beta\nabla_5 \rangle.

    2. \alpha_8 = 0, \alpha_7 = 0, \alpha_6\neq0, then by choosing s = -\frac{\alpha_4}{2\alpha_6}x, we have \alpha_4^* = 0. Consider the following cases:

    {\rm{(a)}}\ \alpha_3 = 0, then we have two families of representatives \langle \alpha\nabla_1+\beta\nabla_2+\nabla_6 \rangle and \langle \alpha\nabla_1+\beta\nabla_2+\nabla_5+\nabla_6 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ \alpha_3\neq0 then by choosing x = \frac{\alpha_6}{\alpha_3}, s = -\frac{\alpha_4}{2\alpha_6}, t = -\frac{\alpha_1\alpha_6}{2\alpha_3^2}, we have the family of representatives \langle \alpha\nabla_2+\nabla_3+\beta\nabla_5+\nabla_6 \rangle.

    3. \alpha_8 = 0, \alpha_7\neq0, then by choosing t = -{\alpha_2}{\alpha_7^{-1}}x, s = -{\alpha_5}{\alpha_7^{-1}}, we have \alpha_2^* = 0, \alpha_5^* = 0. Thus, we can suppose that \alpha_2 = 0, \alpha_5 = 0 and now consider the following cases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_6 = 0, then we have the family of representatives \langle \alpha\nabla_3+\nabla_7 \rangle;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_4 = 0, \alpha_6\neq0, then by choosing x = {\alpha_6}{\alpha_7^{-1}}, s = 0, t = 0, we have the family of representatives \langle \alpha\nabla_3+\nabla_6+\nabla_7 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_4\neq0, then by choosing x = \sqrt[3]{{\alpha_4}{\alpha_7^{-1}}}, s = 0, t = 0, we have the family of representatives \langle \alpha\nabla_3+\nabla_4+\beta\nabla_6+\nabla_7 \rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, then by choosing x = {\alpha_1}{\alpha_7^{-1}}, s = 0, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\nabla_7 \rangle.

    4. \alpha_8\neq0, then by choosing t = -\frac{\alpha_3}{\alpha_8}x, s = -\frac{\alpha_6}{\alpha_8}, we have \alpha_3^* = 0, \alpha_6^* = 0. Thus, we can suppose that \alpha_3 = 0, \alpha_6 = 0. Consider the following cases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_4 = 0, \alpha_5 = 0, then we have the representatives \langle \nabla_8 \rangle and \langle \nabla_7+\nabla_8 \rangle depending on whether \alpha_7 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_4 = 0, \alpha_5\neq0, then by choosing x = \sqrt[3]{{\alpha_5}{\alpha_8^{-1}}}, s = 0, t = 0, we have the family of representatives \langle \nabla_5+\alpha\nabla_7+\nabla_8 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_4\neq0 then by choosing x = \sqrt[4]{{\alpha_4}{\alpha_8^{-1}}}, s = 0, t = 0, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\beta\nabla_7+\nabla_8 \rangle;

    {\rm{(d)}}\ if \alpha_1 = 0, \alpha_2\neq0, then by choosing x = \sqrt{{\alpha_2}{\alpha_8^{-1}}}, s = 0, t = 0, we have the family of representatives \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_7+\nabla_8 \rangle;

    {\rm{(e)}}\ if \alpha_1\neq0, then by choosing x = \sqrt{{\alpha_1}{\alpha_8}^{-1}}, s = 0, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_7+\nabla_8 \rangle .

    Summarizing all cases, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_7+\nabla_8 \rangle ^{ { \begin{array}{l}O(\alpha,\beta,\gamma,\mu) = O(-\alpha,-\beta,-\gamma,\mu) = \\ O(-\alpha,\beta,\gamma,-\mu) = O(\alpha,-\beta,-\gamma,-\mu) \end{array}}}, \\ \langle \alpha\nabla_1+\beta\nabla_2+\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\alpha,\beta)}, \langle \alpha\nabla_1+\beta\nabla_2+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\alpha,\beta)}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_6+\nabla_7 \rangle^{O(\alpha,\beta,\gamma) = O(-\alpha,\beta,-\gamma)}, \langle \nabla_2+\nabla_3+\alpha\nabla_4+\beta\nabla_5 \rangle^{O(\alpha,\beta) = O(-\alpha,\beta)}, \langle \alpha\nabla_2+\nabla_3 +\beta\nabla_5+\nabla_6 \rangle, \\ \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_7+\nabla_8 \rangle^{O(\alpha,\beta,\gamma) = O(\alpha,i\beta, i\gamma) = O(\alpha,-i\beta, -i\gamma)}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+\nabla_4+\alpha\nabla_5 \rangle ^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \\ \langle \alpha\nabla_3+\nabla_4+\beta\nabla_6+\nabla_7 \rangle^{{ \begin{array}{l} O(\alpha,\beta) = O(-\alpha,-\beta) = O(-\alpha,\eta_3\beta) = \\O(-\alpha,-\eta_3^2\beta) = O(\alpha,-\eta_3\beta) = O(\alpha,\eta_3^2\beta)\end{array}}}, \langle \nabla_3+\nabla_5 \rangle, \\ \langle \alpha\nabla_3+\nabla_6+\nabla_7 \rangle ^{O(\alpha,\beta) = O(-\alpha,\beta)}, \langle \alpha\nabla_3+\nabla_7 \rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_4+\alpha\nabla_5+\beta\nabla_7+\nabla_8 \rangle ^{ { \begin{array}{l}O(\alpha,\beta) = O(i\alpha,-i\beta) = O(-i\alpha,i\beta) = O(-\alpha,-\beta) = \\ O(\alpha,-\beta) = O(i\alpha,i\beta) = O(-i\alpha,-i\beta) = O(-\alpha,\beta) \end{array}}}, \\ \langle \nabla_5+\alpha\nabla_7+\nabla_8 \rangle^{ O(\alpha) = O(\eta_3\alpha) = O(-\eta_3^2\alpha) = O(-\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_7+\nabla_8 \rangle, \langle \nabla_8\rangle, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{271}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = \beta e_5 \\ && e_2e_3 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{272}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{273}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{274}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{275}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{276}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{277}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{278} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{279}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{280}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{281} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{282}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{283}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{284}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{285}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{286} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{287} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{19}^{4*}:

    {\mathbf{N}}^{4*}_{19} \begin{array}{ll} e_1e_1 = e_4 & e_1e_2 = e_3 \\ e_2e_2 = e_4 & e_3e_3 = e_4 \end{array} \begin{array}{l} \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{19}) = \langle [\Delta_{11}],[\Delta_{13}],[\Delta_{22}],[\Delta_{23}]\rangle\\ \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{19}) = \mathrm{H}^2_{\mathfrak{D}}(\mathbf{N}^{4*}_{19})\oplus \langle [\Delta_{14}], [\Delta_{24}], [\Delta_{34}], [\Delta_{44}] \rangle \end{array}
    { \begin{array}{l} \begin{array}{l} \phi_1 = \begin{pmatrix} x&0&0&0\\ 0 & q&0&0\\ 0&0 & xq&0\\ t & s&0&1 \end{pmatrix}, \\ {x^2 = 1,q^2 = 1} \end{array} \begin{array}{l} \phi_2 = \begin{pmatrix} 0 & p&0&0\\ y&0&0&0\\ 0&0 & yp&0\\ t & s&0&1 \end{pmatrix}, \\ {y^2 = 1, p^2 = 1} \end{array} \end{array} }

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    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{11}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{22}], \\ \nabla_5 = [\Delta_{23}], & \nabla_6 = [\Delta_{24}], & \nabla_7 = [\Delta_{34}], & \nabla_8 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{8}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4*}_{19}) . Since

    \phi^T\begin{pmatrix} \alpha_1&0&\alpha_2&\alpha_3\\ 0&\alpha_4&\alpha_5&\alpha_6\\ \alpha_2&\alpha_5&0&\alpha_7\\ \alpha_3&\alpha_6&\alpha_7&\alpha_8 \end{pmatrix}\phi = \begin{pmatrix} \alpha_1^*&\alpha^{*}&\alpha^{*}_2&\alpha_3^*\\ \alpha^{*}&\alpha^*_4&\alpha^*_5&\alpha_6^*\\ \alpha^{*}_2&\alpha^*_5&0&\alpha^*_7\\ \alpha^*_3&\alpha^*_6&\alpha^*_7&\alpha^*_8 \end{pmatrix},

    then, in the case \phi = \phi_1^{x = 1,q = 1}, we have

    \begin{array}{llll} \alpha_1^* = \alpha_1+2\alpha_3t+\alpha_8t^2, & \alpha_2^* = \alpha_2+\alpha_7t, & \alpha_3^* = \alpha_3+\alpha_8t, & \alpha_4^* = \alpha_4+2\alpha_6s+\alpha_8s^2, \\ \alpha_5^* = \alpha_5+\alpha_7s, & \alpha_6^* = \alpha_6+\alpha_8s, & \alpha_7^* = \alpha_7, & \alpha_8^* = \alpha_8. \end{array}

    For define the main families of representatives, we will use \phi = \phi_1^{x = 1,q = 1} and for find equal orbits we will use other automorphisms. We are interested in

    (\alpha_3,\alpha_6,\alpha_7,\alpha_8)\neq(0,0,0,0) .

    Let us consider the following cases:

    1.\ if \alpha_8 = 0, \alpha_7 = 0, \alpha_6 = 0, then \alpha_3\neq0 and choosing t = -\frac{\alpha_1}{2\alpha_3}, we have the family of representatives \langle \alpha\nabla_2+\nabla_3+\beta\nabla_4+\gamma\nabla_5 \rangle;

    2.\ if \alpha_8 = 0, \alpha_7 = 0, \alpha_6\neq0 and \alpha_3 = 0, then by choosing some suitable automorphism \phi_2 we have \alpha_3^*\neq0 , \alpha_6^* = 0, which is the case considered above;

    3.\ if \alpha_8 = 0, \alpha_7 = 0, \alpha_6\neq0, \alpha_3\neq0, then by choosing t = -\frac{\alpha_1}{2\alpha_3}, s = \frac{\alpha_4}{2\alpha_6}, we have the family of representatives \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_5+\nabla_6 \rangle_{\beta\neq0} ;

    4.\ if \alpha_8 = 0, \alpha_7\neq0, then by choosing t = -{\alpha_2}{\alpha_7}^{-1}, s = -{\alpha_5}{\alpha_7}^{-1}, we have the family of representatives \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_6+\nabla_7 \rangle;

    5.\ if \alpha_8\neq0, then by choosing t = -{\alpha_3}{\alpha_8}^{-1}, s = -{\alpha_6}{\alpha_8}^{-1}, we have the family of representatives \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_7+\nabla_8 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_7+\nabla_8 \rangle ^{{ \begin{array}{l} O(\alpha,\beta,\gamma,\mu,\nu) = O(\alpha,-\beta,\gamma,\mu,-\nu) = \\ O(\alpha,\beta,\gamma,-\mu,-\nu) = O(\alpha,-\beta,\gamma,-\mu,\nu) = \\ O(\gamma,\mu,\alpha,\beta,\nu) = O(\gamma,-\mu,\alpha,\beta,-\nu) = \\ O(\gamma,\mu,\alpha,-\beta,-\nu) = O(\gamma,-\mu,\alpha,-\beta,\nu) \end{array}}}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_6+\nabla_7 \rangle ^{{ \begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\alpha,-\beta,-\gamma,\mu) = \\ O(-\alpha,\beta,-\gamma,-\mu) = O(\alpha,-\beta,\gamma,-\mu) = \\ O(\gamma,\mu,\alpha,\beta) = O(-\gamma,-\mu,-\alpha,\beta) = \\ O(-\gamma,\mu,-\alpha,-\beta) = O(\gamma,-\mu,\alpha,-\beta) \end{array}}}, \\ \langle \alpha\nabla_2+\nabla_3+\beta\nabla_4+\gamma\nabla_5 \rangle^{{ \begin{array}{l} O(\alpha,\beta,\gamma) = O(-\alpha,\beta,\gamma) = O(-\alpha,-\beta,\gamma) = O(\alpha,-\beta,\gamma) = \end{array}}}, \\ \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_5+\nabla_6 \rangle_{\beta \neq0}^{{ \begin{array}{l} O(\alpha,\beta,\gamma) = O(\alpha,-\beta,-\gamma) = O(\frac{\gamma}{\beta},\frac{1}{\beta},\frac{\alpha}{\beta}) = O(\frac{\gamma}{\beta},-\frac{1}{\beta},-\frac{\alpha}{\beta}) \end{array}}},\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{288}^{\alpha, \beta, \gamma,\mu,\nu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 & e_2e_2 = e_4+\gamma e_5 \\ & & e_2e_3 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{289}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_4 = \beta e_5 & e_2e_2 = e_4+\gamma e_5 \\ & & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{290}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{291}^{\alpha, \beta\neq0 ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{01}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{01} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_2e_3 = e_4 \end{array} &\begin{array}{l}\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{01}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(1,2),(2,3)\} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0 & x^2&0&0\\ z&0 & x^3&0\\ t&0 & x^2z & x^5 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{01}) . Since

    \phi^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&\alpha_3&0&\alpha_4\\ \alpha_1&0&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&\alpha^*_3&\alpha^{***}&\alpha^*_4\\ \alpha^{*}_1&\alpha^{***}&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{lll} { \alpha_1^* = \big((\alpha_1x+\alpha_5z+\alpha_6t)x+(\alpha_2x+\alpha_6z+\alpha_7t)z\big)x^2, }\\ \alpha_2^* = (\alpha_2x+\alpha_6z+\alpha_7t)x^5, & \alpha_3^* = \alpha_3x^4, & \alpha_4^* = \alpha_4x^7, \\ \alpha_5^* = (\alpha_5x^{2}+2\alpha_6xz+\alpha_7z^2)x^4, & \alpha_6^* = (\alpha_6x+\alpha_7z)x^{7}, & \alpha_7^* = \alpha_7x^{10}. \end{array}

    We are interested in (\alpha_2,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0) and consider following cases:

    1.\ \alpha_7 = \alpha_6 = \alpha_4 = 0, then \alpha_2\neq0, and we have the following subcases:

    {\rm{(a)}}\ if \alpha_5 = -\alpha_2, then we have

    {\rm{(i)}}\ if \alpha_1 = 0, \alpha_3 = 0, then we have the representative \langle \nabla_2-\nabla_5 \rangle;

    {\rm{(ii)}}\ if \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \sqrt{{\alpha_3}{\alpha_2}^{-1}}, z = 0, t = 0, we have the representative \langle \nabla_2+\nabla_3-\nabla_5 \rangle;

    {\rm{(iii)}}\ if \alpha_1\neq0, then by choosing x = \sqrt{{\alpha_1}{\alpha_2}^{-1}}, z = 0, t = 0, we have the family of representatives \langle \nabla_1+\nabla_2+\alpha\nabla_3-\nabla_5 \rangle;

    (b) if \alpha_5\neq-\alpha_2, then by choosing z = -\frac{\alpha_1}{\alpha_5+\alpha_2}x, t = 0, we have the family of representatives \langle \nabla_2+\alpha\nabla_5 \rangle_{\alpha\neq-1} and \langle \nabla_2+\nabla_3+\alpha\nabla_5 \rangle_{\alpha\neq-1} depending on whether \alpha_3 = 0 or not, which will be jointed with the cases (1(a)i) and (1(a)ii).

    2. \alpha_7 = 0, \alpha_6 = 0, \alpha_4\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_5 = -\alpha_2, \alpha_1 = 0,

    {\rm{(i)}}\ if \alpha_3 = 0, then we have the representatives \langle \nabla_4 \rangle and \langle \nabla_2+\nabla_4-\nabla_5 \rangle depending on whether \alpha_2 = 0 or not;

    {\rm{(ii)}}\ if \alpha_3\neq0, then by choosing x = \sqrt[3]{{\alpha_3}{\alpha_4}^{-1} }, we have the family of representatives \langle \alpha\nabla_2+\nabla_3+ \nabla_4-\alpha\nabla_5 \rangle;

    (b) if \alpha_5 = -\alpha_2, \alpha_1\neq0, then by choosing x = \sqrt[3]{{\alpha_1}{\alpha_4}^{-1} }, we have the family of representatives \langle \nabla_1+\alpha\nabla_2+\beta\nabla_3+ \nabla_4-\alpha\nabla_5 \rangle;

    (c) if \alpha_5\neq-\alpha_2, then we have

    {\rm{(i)}}\ if \alpha_3 = 0, \alpha_2 = 0, then by choosing x = \frac{\alpha_5}{\alpha_4}, z = -\frac{\alpha_1\alpha_5}{\alpha_4(\alpha_2+\alpha_5)}, t = 0, we have the representative \langle \nabla_4+\nabla_5 \rangle;

    {\rm{(ii)}}\ if \alpha_3 = 0, \alpha_2\neq0, then by choosing x = \frac{\alpha_2}{\alpha_4}, z = -\frac{\alpha_1\alpha_2}{\alpha_4(\alpha_2+\alpha_5)}, t = 0, we have the family of representatives \langle \nabla_2+\nabla_4+\alpha\nabla_5 \rangle_{\alpha\neq-1}, which will be jointed with a representative from the case (2(a)i);

    {\rm{(iii)}}\ if \alpha_3\neq0, then by choosing x = \sqrt[3]{{\alpha_3}{\alpha_4}^{-1}}, z = -\frac{\alpha_1\sqrt[3]{\alpha_3}}{(\alpha_2+\alpha_5)\sqrt[3]{\alpha_4}}, t = 0, we have the family of representatives \langle \alpha\nabla_2+\nabla_3+\nabla_4+\beta\nabla_5 \rangle_{\beta\neq-\alpha}, which will be jointed with the family from the case (2(a)i).

    3. \alpha_7 = 0, \alpha_6\neq0, then we consider the following subcases:

    {\rm{(a)}}\ if \alpha_3 = 0, \alpha_4 = 0, then choosing z = -\frac{\alpha_2}{\alpha_6}x, t = -\frac{\alpha_1x+\alpha_5z}{\alpha_6}, we have representatives \langle \nabla_6 \rangle and \langle \nabla_5+\nabla_6 \rangle depending on whether \alpha_5 = 2\alpha_2 or not;

    {\rm{(b)}}\ if \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \frac{\alpha_4}{\alpha_6}, z = -\frac{\alpha_2\alpha_4}{\alpha^2_6}, t = \frac{\alpha_4(\alpha_2\alpha_5-\alpha_1\alpha_6)}{\alpha^3_6}, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle;

    {\rm{(c)}}\ if \alpha_3\neq0, then by choosing x = \sqrt[4]{\frac{\alpha_3}{\alpha_6}}, z = -\frac{\alpha_2\sqrt[4]{\alpha_3}}{\alpha_6\sqrt[4]{\alpha_6}}, t = \frac{(\alpha_2\alpha_5-\alpha_1\alpha_6)\sqrt[4]{\alpha_3}}{\alpha_6^2\sqrt[4]{\alpha_6}}, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle.

    4. \alpha_7\neq0, then by choosing z = -\frac{\alpha_6}{\alpha_7}x, t = \frac{\alpha_6^2+\alpha_2\alpha_7}{\alpha^2_7}x, we have \alpha_2^* = 0, \alpha_6^* = 0. Thus, we can suppose that \alpha_2 = 0, \alpha_6 = 0 and now consider following subcases:

    {\rm{(a)}}\ \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, then we have representatives \langle \nabla_7 \rangle and \langle \nabla_5+\nabla_7 \rangle depending on whether \alpha_5\alpha_7-\alpha^2_6 = 0 or not;

    {\rm{(b)}}\ \alpha_1 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[3]{{\alpha_4}{\alpha_7}^{-1}}, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\nabla_7 \rangle;

    {\rm{(c)}}\ \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \sqrt[6]{{\alpha_3}{\alpha_7}^{-1}}, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_7 \rangle;

    {\rm{(d)}}\ \alpha_1\neq0, then by choosing x = \sqrt[6]{{\alpha_1}{\alpha_7^{-3}}}, we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_7 \rangle.

    Summarizing all cases, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_3 + \nabla_4 -\alpha \nabla_5 \rangle^ {O(\alpha, \beta) = O(-\eta_3\alpha, \eta_3 \beta) = O(\eta_3^2\alpha,-\eta_3^2\beta)}, \\ \langle \nabla_1+ \nabla_2 + \alpha \nabla_3 - \nabla_5 \rangle, \\ \langle \nabla_1+ \alpha \nabla_3 + \beta \nabla_4 + \gamma \nabla_5 + \nabla_7 \rangle^ {{ \begin{array}{l} O(\alpha, \beta, \gamma) = O(\alpha, \beta, -\eta_3\gamma) = O(\alpha, -\beta, -\eta_3\gamma) = \\ O(\alpha, -\beta, \eta_3^2 \gamma) = O(\alpha, \beta, \eta_3^2\gamma) = O(\alpha, -\beta, \gamma) \end{array} }}, \\ \langle \alpha \nabla_2 + \nabla_3 + \nabla_4 +\beta \nabla_5 \rangle ^{O(\alpha, \beta) = O(-\eta_3\alpha, -\eta_3 \beta) = O(\eta_3^2\alpha, \eta_3^2\beta)}, \langle \nabla_2+ \nabla_3 + \alpha \nabla_5 \rangle, \\ \langle \nabla_2+ \nabla_4 + \alpha \nabla_5 \rangle, \langle \nabla_2 + \alpha \nabla_5 \rangle, \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle ^{{ \begin{array}{l} O(\alpha, \beta) = O(-i\alpha, -\beta) = \\ O(i\alpha, -\beta) = O(-\alpha, \beta) \end{array}}}, \\ \langle \nabla_3 + \alpha\nabla_4 + \beta \nabla_5 + \nabla_7 \rangle^ {{ \begin{array}{l} O(\alpha, \beta) = O(\alpha, -\eta_3 \beta) = O(-\alpha, -\eta_3\beta) = \\ O(-\alpha, \eta_3^2 \beta) = O(\alpha, \eta_3^2 \beta) = O(-\alpha, \beta) \end{array}}}, \langle \nabla_4 \rangle, \langle \nabla_4 + \nabla_5 \rangle, \\ \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle, \langle \nabla_4 + \alpha \nabla_5 + \nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_5+\nabla_6 \rangle, \langle \nabla_5 + \nabla_7 \rangle, \\ \langle \nabla_6 \rangle, \langle \nabla_7 \rangle. \end{array}

    Hence, we have the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{292}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = \beta e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = -\alpha e_5 \\ {\mathbf{N}}_{293}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{294}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{295}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{296}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{297}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{298}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{299}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{300}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{301} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{302} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{303}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{304}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{305} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{306} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{307} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{308} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{02}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{02} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_1e_3 = e_4 \\ e_2e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{02}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(1,2),(1,3)\} \end{array} & \phi = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ z&0&1&0\\ t&2z&z&1 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{14}], & \nabla_2 = [\Delta_{22}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{02}) . Since

    \phi^T\begin{pmatrix} 0&0&0&\alpha_1\\ 0&\alpha_2&\alpha_3&\alpha_4\\ 0&\alpha_3&\alpha_5&\alpha_6\\ \alpha_1&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{***}&\alpha^*_1\\ \alpha^{**}&\alpha^*_2&\alpha^*_3+\alpha^{***}&\alpha^*_4\\ \alpha^{***}&\alpha^*_3+\alpha^{***}&\alpha^*_5&\alpha^*_6\\ \alpha^*_1&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{lcl} \alpha_1^* & = & \alpha_1+\alpha_6z+\alpha_7t, \\ \alpha_2^* & = & \alpha_2+4\alpha_4z+4\alpha_7z^2, \\ \alpha_3^* & = & \alpha_3+2\alpha_6z+(\alpha_4+2\alpha_7z)z-(\alpha_5z+\alpha_6t)-(\alpha_1+\alpha_6z+\alpha_7t)z, \\ \alpha_4^* & = & \alpha_4+2\alpha_7z, \\ \alpha_5^* & = & \alpha_5+2\alpha_6z+\alpha_7z^2, \\ \alpha_6^* & = & \alpha_6+\alpha_7z, \\ \alpha_7^* & = & \alpha_7. \end{array}

    We are interested in (\alpha_1,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0) and consider following cases:

    1.\ if \alpha_7 = \alpha_6 = \alpha_4 = 0, then \alpha_1\neq0, and we have

    {\rm{(a)}}\ if \alpha_5 = -\alpha_1 , then we have the family of representatives

    \langle \nabla_1+\alpha\nabla_2+\beta\nabla_3-\nabla_5 \rangle;

    {\rm{(b)}}\ if \alpha_5\neq-\alpha_1 , then by choosing z = -\frac{\alpha_3}{\alpha_1+\alpha_5}, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_2+\beta\nabla_5 \rangle_{\beta\neq-1};

    2. if \alpha_7 = 0, \alpha_6 = 0, \alpha_4\neq0, then by choosing z = -\frac{\alpha_2}{4\alpha_4}, t = 0, we have the family of representatives \langle \alpha\nabla_1+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle;

    3. if \alpha_7 = 0, \alpha_6\neq0, then by choosing

    z = -{\alpha_1}{\alpha_6}^{-1}, t = ({\alpha_3\alpha_6-\alpha_1(2\alpha_6+\alpha_4-\alpha_5)}){\alpha_6^{-1}},

    we have the family of representatives \langle \alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle;

    4. if \alpha_7\neq0, then by choosing z = -{\alpha_6}{\alpha_7}^{-1}, t = ({\alpha^2_6-\alpha_1\alpha_7}){\alpha^{-2}_7}, we have the family of representatives \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_3 - \nabla_5 \rangle, \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_5 \rangle_{\beta\neq -1}, \langle \alpha \nabla_1+ \beta \nabla_3 + \nabla_4 + \gamma\nabla_5 \rangle, \\ \langle \alpha \nabla_2 + \beta \nabla_3 + \gamma\nabla_4 + \mu \nabla_5 + \nabla_7\rangle, \langle \alpha \nabla_2 + \beta \nabla_4 +\gamma\nabla_5+ \nabla_6\rangle, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{309}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = e_4+\beta e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{310}^{\alpha, \beta\neq-1} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{311}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{312}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{313}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{03}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{03} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_3e_3 = e_4 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{03}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ {(i,j) \notin \{ (1,1),(1,2),(3,3)}\} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0&x^3&0\\ t&0&0&x^6 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{03}) . Since

    \phi^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ \alpha_1&\alpha_4&0&\alpha_6\\ \alpha_2&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&\alpha^*_3&\alpha^{*}_4&\alpha^*_5\\ \alpha^{*}_1&\alpha^{*}_4&0&\alpha^*_6\\ \alpha^*_2&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{llll} \alpha_1^* = (\alpha_1x+\alpha_6t)x^3, & \alpha_2^* = (\alpha_2x+\alpha_7t)x^6, & \alpha_3^* = \alpha_3x^4, & \alpha_4^* = \alpha_4x^5, \\ \alpha_5^* = \alpha_5x^8, & \alpha_6^* = \alpha_6x^{9}, & \alpha_7^* = \alpha_7x^{12}. \end{array}

    We are interested in (\alpha_2,\alpha_5,\alpha_6,\alpha_7)\neq(0,0,0,0) and consider following cases:

    1.\ \alpha_7 = \alpha_6 = \alpha_5 = 0, then \alpha_2\neq0, and we have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_3 = 0, then we have the representatives \langle \nabla_2 \rangle and \langle \nabla_2+\nabla_4 \rangle depending on whether \alpha_4 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_3\neq0, then by choosing x = \sqrt[3]{{\alpha_3}{\alpha_2}^{-1}}, t = 0, we have the family of representatives \langle \nabla_2+\nabla_3+\alpha\nabla_4 \rangle;

    {\rm{(c)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[3]{{\alpha_1}{\alpha_2^{-1}}}, t = 0, we have the family of representatives \langle \nabla_1+\nabla_2+\alpha\nabla_3+\beta\nabla_4 \rangle.

    2. \alpha_7 = 0, \alpha_6 = 0, \alpha_5\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_3 = 0, then we have the representatives \langle \nabla_5 \rangle and \langle \nabla_4+\nabla_5 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_2 = 0, \alpha_3\neq0, then by choosing x = \sqrt[4]{{\alpha_3}{\alpha_5^{-1}}}, t = 0, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_1 = 0, \alpha_2\neq0, then by choosing x = {\alpha_2}{\alpha_5^{-1}}, t = 0, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+\beta\nabla_4+\nabla_5 \rangle;

    {\rm{(d)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[4]{{\alpha_1}{\alpha_5^{-1}}}, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\nabla_5 \rangle.

    3. \alpha_7 = 0, \alpha_6\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_3 = 0, \alpha_4 = 0, then we have representatives \langle \nabla_6 \rangle and \langle \nabla_5+\nabla_6 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[4]{{\alpha_4}{\alpha_6}^{-1}}, t = -\alpha_1 \sqrt[4]{\alpha_4 \alpha_6^{-5}} , we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle;

    {\rm{(c)}}\ if \alpha_2 = 0, \alpha_3\neq0, then by choosing x = \sqrt[5]{{\alpha_3}{\alpha_6^{-1}}}, t = - \alpha_1\sqrt[5]{\alpha_3 \alpha_6^{-6}}, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle;

    {\rm{(d)}}\ if \alpha_2\neq0, then by choosing x = {{\alpha_2}{\alpha_6^{-1}}}, t = -{\alpha_1\sqrt{\alpha_2 \alpha_6^{-3}}}, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle.

    4. \alpha_7\neq0, then we have the following subcases:

    {\rm{(a)}}\ \alpha_1\alpha_7-\alpha_2\alpha_6 = 0, \alpha_3 = 0, \alpha_4 = 0, \alpha_5 = 0, then we have representatives \langle \nabla_7 \rangle and \langle \nabla_6+\nabla_7 \rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ \alpha_1\alpha_7-\alpha_2\alpha_6 = 0, \alpha_3 = 0, \alpha_4 = 0, \alpha_5\neq0, then by choosing x = \sqrt[4]{{\alpha_5}{\alpha_7^{-1}}}, t = -{\alpha_2\sqrt[4]{\alpha_5 \alpha_7^{-5}}}, we have the family of representatives \langle \nabla_5+\alpha\nabla_6+\nabla_7 \rangle;

    {\rm{(c)}}\ \alpha_1\alpha_7-\alpha_2\alpha_6 = 0, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[7]{{\alpha_4\alpha_7^{-1}}}, t = -{\alpha_2\sqrt[7]{\alpha_4 \alpha_7^{-8}}}, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle;

    {\rm{(d)}}\ \alpha_1\alpha_7-\alpha_2\alpha_6 = 0, \alpha_3\neq0, then by choosing x = \sqrt[8]{{\alpha_3}{\alpha_7^{-1}}}, t = -{\alpha_2\sqrt[8]{\alpha_3 \alpha_7^{-9}}}, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle;

    {\rm{(e)}}\ \alpha_1\alpha_7-\alpha_2\alpha_6\neq0, then by choosing

    x = \sqrt[8]{{(\alpha_1\alpha_7-\alpha_2\alpha_6)}{\alpha^{-2}_7}}, t = -{\alpha_2\sqrt[8]{(\alpha_1\alpha_7-\alpha_2\alpha_6)\alpha_7^{-10}}},

    we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\nabla_2 + \alpha \nabla_3 + \beta \nabla_4 \rangle ^{O(\alpha, \beta) = O(\alpha, -\eta_3 \beta) = O(\alpha, \eta_3^2\beta)}, \\ \langle \nabla_1+ \alpha \nabla_2 + \beta \nabla_3 + \gamma \nabla_4 + \nabla_5\rangle ^{O(\alpha, \beta, \gamma) = O(-i\alpha, \beta, i\gamma) = O(i\alpha, \beta,-i \gamma) = O(-\alpha, \beta, -\gamma)}, \\ \langle \nabla_1+\alpha \nabla_3+ \beta \nabla_4 + \gamma\nabla_5 + \mu\nabla_6 + \nabla_7\rangle^{ {\begin{array}{l} O(\alpha,\beta,\gamma, \mu) = O(\alpha,\eta_4^3\beta,-\gamma, -\eta_4^3\mu) = \\ O(\alpha,-\eta_4^3\beta,-\gamma, \eta_4^3\mu) = O(\alpha,\eta_4 \beta,-\gamma, -\eta_4\mu) = \\ O(\alpha,-\eta_4\beta,-\gamma, \eta_4\mu) = O(\alpha,i\beta,\gamma, i\mu) = \\ O(\alpha,-i\beta,\gamma, -i\mu) = O(\alpha,-\beta, \gamma, -\mu) \end{array}} }, \\ \langle \nabla_2\rangle, \langle \nabla_2+\nabla_3 +\alpha\nabla_4\rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta_3^2\alpha)}, \langle \nabla_2+\alpha\nabla_3 +\beta\nabla_4 + \nabla_5\rangle, \\ \langle \nabla_2+\alpha \nabla_3+ \beta \nabla_4 + \gamma\nabla_5 + \nabla_6\rangle^{O(\alpha,\beta,\gamma) = O(-\alpha,\beta,-\gamma)}, \langle \nabla_2+ \nabla_4\rangle, \\ \langle \nabla_3+\alpha \nabla_4+\nabla_5\rangle^{ { \begin{array}{l} O(\alpha) = O(-\alpha) = \\ O(i\alpha) = O(-i\alpha) \end{array}}}, \\ \langle \nabla_3+\alpha \nabla_4+ \beta \nabla_5+ \nabla_6\rangle ^{{ \begin{array}{l} O(\alpha, \beta) = O(\eta_5^4\alpha, -\eta_5\beta) = O(-\eta_5^3\alpha, \eta_5^2\beta) = \\ O(\eta_5^2\alpha, -\eta_5^3\beta) = O(-\eta_5\alpha, \eta_5^4\beta) \end{array}}}, \\ \langle \nabla_3+\alpha \nabla_4+ \beta \nabla_5 +\gamma\nabla_6 + \nabla_7\rangle^{ {\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_4^3\alpha,-\beta,-\eta_4^3\gamma) = O(-\eta_4^3\alpha,-\beta,\eta_4^3\gamma) = \\ O(\eta_4\alpha,-\beta,-\eta_4\gamma) = O(-\eta_4\alpha,-\beta,\eta_4\gamma) = \\ O(i\alpha,\beta,i\gamma) = O(-i\alpha,\beta,-i\gamma) = O(-\alpha,\beta,-\gamma) \end{array}} }, \\ \langle \nabla_4+ \nabla_5\rangle, \langle \nabla_4+\alpha \nabla_5+ \nabla_6\rangle^{O(\alpha) = O(i\alpha) = O(-\alpha) = O(-i\alpha)}, \\ \langle \nabla_4+\alpha \nabla_5+ \beta\nabla_6 + \nabla_7\rangle^{ {\begin{array}{l} O(\alpha,\beta) = O(\eta^4_7\alpha,-\eta^3_7\beta) = O(-\eta_7\alpha,\eta_7^6\beta) = O(-\eta_7^5\alpha,\eta^2_7\beta) = \\ O(\eta^2_7\alpha,-\eta^5_7\beta) = O(\eta_7^6\alpha,-\eta_7\beta) = O(-\eta^3_7\alpha,\eta^4_7\beta) \end{array}}}, \langle \nabla_5\rangle, \\ \langle \nabla_5 + \nabla_6\rangle, \langle \nabla_5+\alpha \nabla_6+ \nabla_7\rangle^{O(\alpha) = O(i\alpha) = O(-\alpha) = O(-i\alpha)}, \langle \nabla_6\rangle, \langle \nabla_6 +\nabla_7\rangle, \langle \nabla_7\rangle, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{314}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{315}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = \beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{316}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{317} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{318}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{319}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{320}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{321} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{322}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{323}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{324}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{325} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{326}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{327}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{328} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{329} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}^\alpha_{330} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{331} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{332} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{333} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{04}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{04} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_3 \\ e_2e_2 = e_4 \\ e_3e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{04}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{(1,1),(1,2),(3,3)\} \end{array} & \phi_{\pm} = \begin{pmatrix} \pm1&0&0&0\\ 0&1&0&0\\ 0&0&\pm1&0\\ t&0&0&1 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{22}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{04}) . Since

    \phi^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&\alpha_3&\alpha_4&\alpha_5\\ \alpha_1&\alpha_4&0&\alpha_6\\ \alpha_2&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&\alpha^*_3&\alpha^{*}_4&\alpha^*_5\\ \alpha^{*}_1&\alpha^{*}_4&0&\alpha^*_6\\ \alpha^*_2&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{llll} \alpha_1^* = \alpha_1\pm\alpha_6t, & \alpha_2^* = \pm\alpha_2+\alpha_7t, & \alpha_3^* = \alpha_3, & \alpha_4^* = \pm\alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \pm \alpha_6, & \alpha_7^* = \alpha_7. \end{array}

    We are interested in (\alpha_2,\alpha_5,\alpha_6,\alpha_7)\neq(0,0,0,0) and consider following cases:

    1.\ if \alpha_7 = \alpha_6 = \alpha_5 = 0, then \alpha_2\neq0, and we have the family of representatives

    \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_4 \rangle;

    2.\ if \alpha_7 = 0, \alpha_6 = 0, \alpha_5\neq0, then we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\mu\nabla_4+\nabla_5 \rangle;

    3.\ if \alpha_7 = 0, \alpha_6\neq0, then by choosing \phi = \phi_+, t = -{\alpha_1}{\alpha_6^{-1}}, we have the family of representatives

    \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle;

    4.\ if \alpha_7\neq0, then by choosing \phi = \phi_+, t = -{\alpha_2}{\alpha_7^{-1}} we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_4 \rangle^{O(\alpha, \beta, \gamma) = O(-\alpha, -\beta, \gamma)}, \\\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\mu\nabla_4+\nabla_5 \rangle ^{O(\alpha, \beta, \gamma,\mu) = O(\alpha, -\beta, \gamma, -\mu)} \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle^{O(\alpha, \beta, \gamma,\mu,\nu) = O(\alpha, \beta, -\gamma,\mu,-\nu)}, \\ \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle ^{O(\alpha, \beta, \gamma, \mu) = O(\alpha, -\beta, \gamma, -\mu)}, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{334}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{335}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_3 = \mu e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{336}^{\alpha, \beta,\gamma, \mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 \\& & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{337}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4+\beta e_5 \\ && e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{05}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms}\\ \hline {\mathbf{N}}^{4}_{05} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_3 = e_4 \\ e_2e_2 = e_3 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{05}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ {(i,j) \notin \{ (1,1),(1,3),(2,2)} \} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ z&0&x^4&0\\ t&2xz&0&x^5 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{12}] & \nabla_2 = [\Delta_{14}] & \nabla_3 = [\Delta_{23}] & \nabla_4 = [\Delta_{24}] \\ \nabla_5 = [\Delta_{33}] & \nabla_6 = [\Delta_{34}] & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{05}) . Since

    \phi^T\begin{pmatrix} 0&\alpha_1&0&\alpha_2\\ \alpha_1&0&\alpha_3&\alpha_4\\ 0&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^*_1&\alpha^{***}&\alpha^*_2\\ \alpha^*_1&\alpha^{**}&\alpha^*_3&\alpha^*_4\\ \alpha^{***}&\alpha^*_3&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{lll} { \alpha_1^* = (\alpha_1x+\alpha_3z+\alpha_4t)x^2+2(\alpha_2x+\alpha_6z+\alpha_7t)xz, }\\ \alpha_2^* = (\alpha_2x+\alpha_6z+\alpha_7t)x^5, & \alpha_3^* = (\alpha_3x+2\alpha_6z)x^5, & \alpha_4^* = (\alpha_4x+2\alpha_7z)x^6, \\ \alpha_5^* = \alpha_5x^8, & \alpha_6^* = \alpha_6x^9, & \alpha_7^* = \alpha_7x^{10}. \end{array}

    We are interested in (\alpha_2,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0) and consider following cases:

    1.\ \alpha_7 = \alpha_6 = \alpha_4 = 0, then \alpha_2\neq0 and we have the following subcases:

    {\rm{(a)}}\ \alpha_3 = -2\alpha_2,

    {\rm{(i)}}\ if \alpha_1 = 0, \alpha_5 = 0, then we have the representative \langle \nabla_2-2\nabla_3 \rangle;

    {\rm{(ii)}}\ if \alpha_1 = 0, \alpha_5\neq0, then by choosing x = \sqrt{{\alpha_2}{\alpha_5^{-1}}}, z = 0, t = 0, we have the representative \langle \nabla_2-2\nabla_3+\nabla_5 \rangle;

    {\rm{(iii)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[3]{{\alpha_1}{\alpha_2^{-1}}}, z = 0, t = 0, we have the family of representatives \langle \nabla_1+\nabla_2-2\nabla_3+\alpha\nabla_5 \rangle.

    (b) \alpha_3\neq -2\alpha_2,

    {\rm{(i)}}\ if \alpha_5 = 0, then choosing x = 1, z = -\frac{\alpha_1}{\alpha_3+2\alpha_2}, we have the family of representatives \langle \nabla_2+\alpha\nabla_3 \rangle_{\alpha\neq-2}, which will be jointed with the case (1(a)i);

    {\rm{(ii)}}\ if \alpha_5\neq0, then by choosing x = \sqrt{\frac{\alpha_2}{\alpha_5}}, z = -\frac{\alpha_1\sqrt{\alpha_2}}{(\alpha_3+2\alpha_2)\sqrt{\alpha_5}}, we have the family of representatives \langle\nabla_2+\alpha\nabla_3+\nabla_5 \rangle_{\alpha\neq-2}, which will be jointed with the case (1(a)ii).

    2. \alpha_7 = \alpha_6 = 0, \alpha_4\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_3 = 0, then we have representatives \langle \nabla_4 \rangle and \langle \nabla_4+\nabla_5 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_3\neq0 , then by choosing x = {\alpha_3}{\alpha_4^{-1}}, z = 0, t = -{\alpha_1\alpha_3}{\alpha_4^{-2}}, we have the family of representatives \langle \nabla_3+ \nabla_4+\alpha\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_2\neq0 , then by choosing x = {\alpha_2}{\alpha_4^{-1}}, z = 0, t = -{\alpha_1\alpha_2}{\alpha_4^{-2}}, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+ \nabla_4+\beta\nabla_5 \rangle.

    3. \alpha_7 = 0, \alpha_6\neq0, then by choosing z = -{\alpha_2}x{\alpha_6^{-1}} , we have \alpha_2^* = 0. Thus, we can suppose that \alpha_2 = 0 and consider following subcases:

    {\rm{(a)}}\ \alpha_4 = 0,

    {\rm{(i)}}\ if \alpha_1 = 0, \alpha_3 = 0, then we have representatives \langle \nabla_6 \rangle and \langle \nabla_5+\nabla_6 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(ii)}}\ if \alpha_1 = 0, \alpha_3\neq0 then by choosing x = \sqrt[3]{{\alpha_3}{\alpha_6^{-1}}}, z = 0, t = 0, we have the family of representatives \langle \nabla_3+\alpha\nabla_5+\nabla_6 \rangle;

    {\rm{(iii)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[6]{{\alpha_1}{\alpha_6^{-1}}}, z = 0, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_5+\nabla_6 \rangle.

    (b) \alpha_4\neq0 , then by choosing x = \sqrt{{\alpha_4}{\alpha_6^{-1}}}, t = -{\alpha_1\sqrt{\alpha_4 \alpha_6^{-3}}}, we have the family of representatives \langle \alpha\nabla_3+\nabla_4+\beta\nabla_5+\nabla_6 \rangle.

    4. \alpha_7\neq0,

    {\rm{(a)}}\ if \alpha_1\alpha_7-\alpha_2\alpha_4 = 0, \alpha_3 = 0, \alpha_6 = 0, then then by choosing z = -\frac{\alpha_4}{2\alpha_7}x, t = \frac{\alpha_4\alpha_6-2\alpha_2\alpha_7}{2\alpha^2_7}x, we have representatives \langle \nabla_7 \rangle and \langle \nabla_5+\nabla_7 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \alpha_1\alpha_7-\alpha_2\alpha_4 = 0, \alpha_3\alpha_7-\alpha_4\alpha_6 = 0, \alpha_6\neq0, then by choosing

    x = \frac{\alpha_6}{\alpha_7}, z = -\frac{\alpha_4\alpha_6}{2\alpha_7^2}, t = \frac{\alpha_6(\alpha_4\alpha_6-2\alpha_2\alpha_7)}{2\alpha^3_7},

    we have the family of representatives \langle \alpha\nabla_5+\nabla_6+\nabla_7 \rangle;

    {\rm{(c)}}\ if 2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7 = 0, \alpha_3\alpha_7-\alpha_4\alpha_6\neq0, then by choosing

    x = \sqrt[4]{\frac{\alpha_3\alpha_7-\alpha_4\alpha_7}{\alpha^2_7}}, z = -\frac{\alpha_4\sqrt[4]{\alpha_3\alpha_7-\alpha_4\alpha_7}}{2\alpha_7\sqrt[4]{2\alpha_7^2}}, t = \frac{(\alpha_4\alpha_6-2\alpha_2\alpha_7)\sqrt[4]{\alpha_3\alpha_7-\alpha_4\alpha_6}}{2\alpha^2_7\sqrt[4]{2\alpha_7^2}},

    we have the family of representatives \langle \nabla_3+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle;

    {\rm{(d)}}\ if 2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7\neq0, then by choosing

    \begin{array}{c} x = \sqrt[7]{\frac{2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7}{2\alpha^3_7}}, \\ z = -\frac{\alpha_4\sqrt[7]{2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7}}{2\alpha_7\sqrt[7]{2\alpha_7^3}}, \\ t = \frac{(\alpha_4\alpha_6-2\alpha_2\alpha_7)\sqrt[7]{2\alpha_1\alpha_7^2-\alpha_3\alpha_4\alpha_7+\alpha_4^2\alpha_6-2\alpha_2\alpha_4\alpha_7}}{2\alpha^3_7\sqrt[7]{2\alpha_7^3}}, \end{array}

    we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\nabla_2-2\nabla_3+\alpha\nabla_5 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta^2_3\alpha)}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_5+\nabla_6 \rangle^{{\begin{array}{l} O(\alpha,\beta) = O(\alpha,-\eta_3\beta) = O(-\alpha,\eta_3\beta) = \\ O(-\alpha,-\eta_3^2\beta) = O(\alpha,\eta_3^2\beta) = O(-\alpha,-\beta) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_7^4\alpha,\eta^2_7\beta,-\eta_7\gamma) = O(-\eta_7\alpha,\eta^4_7\beta,\eta^2_7\gamma) = \\ O(-\eta_7^5\alpha,\eta^6_{7}\beta,-\eta^3_7\gamma) = O(\eta^2_7\alpha,-\eta_7\beta,\eta^4_7\gamma) = \\ O(\eta_7^6\alpha,-\eta^3_7\beta,-\eta^5_7\gamma) = O(-\eta^3_7\alpha,-\eta^5_7\beta,\eta^6_7\gamma) \end{array}}}, \\ \langle \nabla_2+\alpha\nabla_3 \rangle, \langle \nabla_2+\alpha\nabla_3+ \nabla_4+\beta\nabla_5 \rangle, \langle\nabla_2+\alpha\nabla_3+\nabla_5 \rangle, \langle \nabla_3+ \nabla_4+\alpha\nabla_5 \rangle, \\ \langle \alpha\nabla_3+\nabla_4+\beta\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\alpha,-\beta)}, \langle \nabla_3+\alpha\nabla_5+\nabla_6 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta^2_3\alpha)}, \\ \langle \nabla_3+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle^{O(\alpha,\beta) = O(-\alpha,-i\beta) = O(-\alpha,i \beta) = O(\alpha,-\beta)}, \langle \nabla_4 \rangle, \langle \nabla_4+\nabla_5 \rangle, \\ \langle \nabla_5+\nabla_6 \rangle, \langle \alpha\nabla_5+\nabla_6+\nabla_7 \rangle, \langle \nabla_5+\nabla_7 \rangle, \langle\nabla_6 \rangle, \langle\nabla_7 \rangle,\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{338}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{339}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{340}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{341}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{342}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{343}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{344}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{345}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{346}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{347}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{348} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ {\mathbf{N}}_{349} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{350} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{351}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{352} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{353} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{354} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{06}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{06} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_4 \\ e_1e_3 = e_4 \\ e_2e_2 = e_3 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{06}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{(1,1),(1,2),(2,2)\} \end{array} & \phi_{\pm} = \begin{pmatrix} \pm1&0&0&0\\ 0&1&0&0\\ z&0&1&0\\ t&\pm2z&0&\pm1 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{06}). Since

    \phi_{\pm}^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&0&\alpha_3&\alpha_4\\ \alpha_1&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi_{\pm} = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1+\alpha^{**}&\alpha^*_2\\ \alpha^{**}&\alpha^{***}&\alpha^*_3&\alpha^*_4\\ \alpha^{*}_1+\alpha^{**}&\alpha^*_3&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{lll} { \alpha_1^* = \pm \alpha_1- \alpha_3z- \alpha_4t+ \alpha_5z+ \alpha_6t-2 ( \alpha_2\pm \alpha_6z\pm \alpha_7t)z, }\\ \alpha_2^* = \alpha_2\pm\alpha_6z \pm\alpha_7t, & \alpha_3^* = \alpha_3 \pm 2\alpha_6z, & \alpha_4^* = 2\alpha_7z \pm \alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \pm \alpha_6, & \alpha_7^* = \alpha_7. \end{array}

    Since (\alpha_2,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0) and for \phi = \phi_+, we have the following cases:

    1.\ if \alpha_7 = \alpha_6 = \alpha_4 = 0, then \alpha_2\neq0, and we have the following subcase:

    {\rm{(a)}}\ if \alpha_5 = \alpha_3+2\alpha_2, then we have the family of representatives

    \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+(\beta+2)\nabla_5 \rangle;

    {\rm{(b)}}\ if \alpha_5\neq\alpha_3+2\alpha_2, then by choosing z = 0, t = \frac{\alpha_1}{\alpha_3+2\alpha_2-\alpha_5} , we have the family of representative \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5 \rangle_{\beta\neq\alpha+2};

    2. if \alpha_7 = 0, \alpha_6 = 0, \alpha_4\neq0, then by choosing z = 0, t = \frac{\alpha_1}{\alpha_4}, we have the family of representatives \langle \alpha\nabla_2+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle;

    3. if \alpha_7 = 0, \alpha_6\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_6 = \alpha_4, then by choosing z = -{\alpha_2}{\alpha_6^{-1}}, t = 0, we have the family of representatives \langle \alpha\nabla_1+\beta\nabla_3+\nabla_4+\gamma\nabla_5+\nabla_6 \rangle;

    {\rm{(b)}}\ if \alpha_6\neq\alpha_4, then by choosing z = -\frac{\alpha_2}{\alpha_6}, t = \frac{\alpha_1\alpha_6-\alpha_2\alpha_5+\alpha_2\alpha_3}{\alpha_6(\alpha_4-\alpha_6)}, we have the family of representatives \langle \alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle_{\beta\neq1};

    4. if \alpha_7\neq0, then by choosing z = -\frac{\alpha_4}{2\alpha_7}, t = \frac{\alpha_4\alpha_6-2\alpha_2\alpha_7}{2\alpha^2_7}, we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+(\beta+2)\nabla_5 \rangle_{\alpha \neq 0}^{O(\alpha, \beta) = O(-\alpha, \beta)}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\nabla_4+\gamma\nabla_5+\nabla_6 \rangle^{O(\alpha, \beta, \gamma) = O(\alpha, -\beta,-\gamma)}_{\alpha\neq 0}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle^{O(\alpha, \beta, \gamma, \mu) = O(-\alpha, \beta, \gamma, -\mu)}, \langle \nabla_2+\alpha\nabla_3+\beta\nabla_5 \rangle, \\ \langle \alpha\nabla_2+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle^{O(\alpha, \beta, \gamma) = O(-\alpha, -\beta,- \gamma)}, \\ \langle \alpha \nabla_3+\beta \nabla_4+\gamma\nabla_5+\nabla_6 \rangle^{O(\alpha, \beta, \gamma) = O(-\alpha, \beta, -\gamma)}, \end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{355}^{\alpha\neq0, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4+\alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & { e_3e_3 = (\beta+2)e_5 } \\ {\mathbf{N}}_{356}^{\alpha\neq0, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4+\alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{357}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_3 = e_4+\alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 & \\ {\mathbf{N}}_{358}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{359}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 & \\ {\mathbf{N}}_{360}^{\alpha,\beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ \end{array}

    Here we will collect all information about {\mathbf N}_{07}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{07} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_2 = e_3 \\ e_2e_3 = e_4 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{07}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(2,2),(2,3) \} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0&x^4&0\\ t&0&0&x^6 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{07}) . Since

    \phi^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&0&\alpha_4\\ \alpha_2&0&\alpha_5&\alpha_6\\ \alpha_3&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^*_1&\alpha^{*}_2&\alpha^*_3\\ \alpha^*_1&0&0&\alpha^*_4\\ \alpha^{*}_2&0&\alpha^*_5&\alpha^*_6\\ \alpha^*_3&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{llll} \alpha_1^* = (\alpha_1x+\alpha_4t)x^2, & \alpha_2^* = (\alpha_2x+\alpha_6t)x^4, & \alpha_3^* = (\alpha_3x+\alpha_7t)x^6, & \alpha_4^* = \alpha_4x^8, \\ \alpha_5^* = \alpha_5x^8, & \alpha_6^* = \alpha_6x^{10}, & \alpha_7^* = \alpha_7x^{12}. \end{array}

    We are interested in (\alpha_3,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0) and consider following cases:

    1.\ \alpha_7 = \alpha_6 = \alpha_4 = 0, then \alpha_3\neq0, and we have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_2 = 0, then we have representatives \langle \nabla_3 \rangle and \langle \nabla_3+\nabla_5 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_2\neq0, then by choosing x = \sqrt{{\alpha_2}{\alpha_3^{-1}}}, t = 0, we have the family of representatives \langle \nabla_2+\nabla_3+\alpha\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[4]{{\alpha_1}{\alpha_3^{-1}}}, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_2+\nabla_3+\beta\nabla_5 \rangle.

    2. \alpha_7 = 0, \alpha_6 = 0, \alpha_4\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_3 = 0 , then we have the familty of representative \langle \nabla_4+\alpha\nabla_5 \rangle;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_3\neq0 , then by choosing x = {\alpha_3}{\alpha_4^{-1}}, t = -{\alpha_1\alpha_3}{\alpha_4^{-2}}, we have the family of representatives \langle \nabla_3+ \nabla_4+\alpha\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_2\neq0, then by choosing x = \sqrt[3]{{\alpha_2}{\alpha_4^{-1}}}, t = -{\alpha_1\sqrt[3]{\alpha_2 \alpha_4^{-4}}}, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+ \nabla_4+\beta\nabla_5 \rangle.

    3. \alpha_7 = 0, \alpha_6\neq0, then we have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_3 = 0, \alpha_4 = 0, then we have representatives \langle \nabla_6 \rangle and \langle \nabla_5+\nabla_6 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \alpha_1\alpha_6 = \alpha_2\alpha_4, \alpha_3 = 0, \alpha_4\neq0, then choosing x = \sqrt{{\alpha_4}{\alpha_6^{-1}}}, t = -{\alpha_2\sqrt{\alpha_4} \alpha_6^{-3}}, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle;

    {\rm{(c)}}\ if \alpha_1\alpha_6 = \alpha_2\alpha_4, \alpha_3\neq0, then by choosing x = \sqrt[3]{\frac{\alpha_3}{\alpha_6}}, t = -\frac{\alpha_2\sqrt[3]{\alpha_3}}{\alpha_6\sqrt[3]{\alpha_6}}, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle;

    {\rm{(d)}}\ if \alpha_1\alpha_6\neq\alpha_2\alpha_4 , then choosing x = \sqrt[7]{\frac{\alpha_1\alpha_6-\alpha_2\alpha_4}{\alpha^2_6}}, t = -\frac{\alpha_2\sqrt[7]{\alpha_1\alpha_6-\alpha_2\alpha_4}}{\alpha_6\sqrt[7]{\alpha^2_6}}, we have the family of representatives \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle.

    4. \alpha_7\neq0, then we have the following subcases:

    {\rm{(a)}}\ \alpha_1 = 0, \alpha_2\alpha_7 = \alpha_3\alpha_6, \alpha_4 = 0, \alpha_5 = 0, then we have representatives \langle \nabla_7 \rangle and \langle \nabla_6+\nabla_7 \rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ \alpha_1 = 0, \alpha_2\alpha_7 = \alpha_3\alpha_6, \alpha_4 = 0, \alpha_5\neq0, then by choosing x = \sqrt[4]{{\alpha_5}{\alpha_7^{-1}}}, t = -{\alpha_3\sqrt[4]{\alpha_5 \alpha_7^{-5}}}, we have family of representatives \langle \nabla_5+\alpha\nabla_6+\nabla_7 \rangle;

    {\rm{(c)}}\ \alpha_1\alpha_7 = \alpha_3\alpha_4, \alpha_2\alpha_7 = \alpha_3\alpha_6, \alpha_4\neq0, then by choosing x = \sqrt[4]{{\alpha_4}{\alpha_7^{-1}}}, t = -{\alpha_3\sqrt[4]{\alpha_4 \alpha_7^{-5}}}, we have family of representatives \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle;

    {\rm{(d)}}\ \alpha_1\alpha_7 = \alpha_3\alpha_4, \alpha_2\alpha_7\neq\alpha_3\alpha_6, then by choosing x = \sqrt[7]{(\alpha_2\alpha_7-\alpha_3\alpha_6)\alpha^{-2}_7}, t = -{\alpha_3\sqrt[7]{(\alpha_2\alpha_7-\alpha_3\alpha_6)\alpha_7^{-9}}}, we have the family of representatives

    \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle;

    {\rm{(e)}}\ \alpha_1\alpha_7\neq \alpha_3\alpha_4, then by choosing

    x = \sqrt[9]{(\alpha_1\alpha_7-\alpha_3\alpha_4)\alpha^{-2}_7}, t = -\alpha_3\sqrt[9]{(\alpha_1\alpha_7-\alpha_3\alpha_4)\alpha_7^{-11} },

    we have family of representatives \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\nabla_3+\beta\nabla_5 \rangle ^{O(\alpha,\beta) = O(-\alpha,i\beta) = O(-\alpha,-i\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle ^{{\begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\eta_9^7\alpha,\eta^4_9\beta,\eta^4_9\gamma,\eta^2_9\mu) = \\ O(-\eta^5_9\alpha,\eta_9^8\beta,\eta_9^8\gamma,\eta^4_9\mu) = O(-\eta_3\alpha,-\eta_3\beta,-\eta_3\gamma,\eta^2_3\mu) = \\ O(-\eta_9\alpha,-\eta^7_9\beta,-\eta^7_9\gamma,\eta^8_9\mu) = O(\eta_9^8\alpha,\eta^2_9\beta,\eta^2_9\gamma,-\eta_9\mu) = \\ O(\eta^2_3\alpha,\eta^2_3\beta,\eta^2_3\gamma,-\eta_3\mu) = O(\eta^4_9\alpha,-\eta_9\beta,-\eta_9\gamma,-\eta^5_9\mu) = \\ O(\eta^2_9\alpha,-\eta^5_9\beta,-\eta^5_9\gamma,-\eta^7_9\mu) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle ^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(-\eta_7^3\alpha,\eta^2_7\beta,\eta^2_7\gamma) = \\ O(\eta_7^6\alpha,\eta^4_7\beta,\eta^4_7\gamma) = O(\eta^2_7\alpha,\eta^6_{7}\beta,\eta^6_7\gamma) = \\ O(-\eta^5_7\alpha,-\eta_7\beta,-\eta_7\gamma) = O(-\eta_7\alpha,-\eta^3_7\beta,-\eta^3_7\gamma) = \\ O(\eta^4_7\alpha,-\eta^5_7\beta,-\eta^5_7\gamma) \end{array}}}, \\ \langle \nabla_2+\alpha\nabla_3+ \nabla_4+\beta\nabla_5 \rangle^{O(\alpha,\beta) = O(-\eta_3\alpha,\beta) = O(\eta^2_3\alpha,\beta)}, \langle \nabla_2+\nabla_3+\alpha\nabla_5 \rangle^{O(\alpha) = O(-\alpha)}, \\ \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_7^4\alpha,\eta^4_7\beta,\eta^2_7\gamma) = O(-\eta_7\alpha,-\eta_7\beta,\eta^4_7\gamma) = \\ O(-\eta^5_7\alpha,-\eta^5_{7}\beta,\eta^6_7\gamma) = O(\eta^2_7\alpha,\eta^2_7\beta,-\eta_7\gamma) = \\ O(\eta_7^6\alpha,\eta_7^6\beta,-\eta^3_7\gamma) = O(-\eta^3_7\alpha,-\eta^3_7\beta,-\eta^5_7\gamma) \end{array}}}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+ \nabla_4+\alpha\nabla_5 \rangle, \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(-\eta_3\alpha,-\eta_3\beta) = O(\eta^2_3\alpha,\eta^2_3\beta)}, \\ \langle \nabla_3+\nabla_5 \rangle, \langle \nabla_4+\alpha\nabla_5 \rangle, \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle, \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle^{O(\alpha,\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_5+\nabla_6 \rangle, \langle \nabla_5+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\alpha) }, \langle \nabla_6 \rangle, \langle \nabla_6 +\nabla_7 \rangle, \langle \nabla_7 \rangle,\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{361}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{362}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{363}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{364}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{365}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{366}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{367} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ {\mathbf{N}}_{368}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{369}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{370} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{371}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{372}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{373}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ & & e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{374} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{375}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{376} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{377} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{378} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{08}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms}\\ \hline {\mathbf{N}}^{4}_{08} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_3 = e_4 \\ e_2e_2 = e_3 \\ e_2e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{08}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ {(i,j) \notin \{(1,1),(1,3),(2,2)}\} \end{array} & \phi = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ t&0&0&1 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{08}) . Since

    \phi^T\begin{pmatrix} 0&\alpha_1&0&\alpha_2\\ \alpha_1&0&\alpha_3&\alpha_4\\ 0&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^{*}_1&\alpha^{**}&\alpha^*_2\\ \alpha^{*}_1&0&\alpha^*_3+\alpha^{**}&\alpha^*_4\\ \alpha^{**}&\alpha^*_3+\alpha^{**}&\alpha^*_5&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{llll} \alpha_1^* = \alpha_1+\alpha_4t, & \alpha_2^* = \alpha_2+\alpha_7t, & \alpha_3^* = \alpha_3-\alpha_6t, & \alpha_4^* = \alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \alpha_6, & \alpha_7^* = \alpha_7. \end{array}

    Since (\alpha_2,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0), we have the following cases:

    1.\ if \alpha_7 = \alpha_6 = \alpha_4 = 0, then \alpha_2\neq0, and we have the family of representatives

    \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5 \rangle;

    2.\ if \alpha_7 = 0, \alpha_6 = 0, \alpha_4\neq0, then by choosing t = -{\alpha_1}{\alpha_4^{-1}}, we have the family of representatives

    \langle \alpha\nabla_2+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle;

    3.\ if \alpha_7 = 0, \alpha_6\neq0, then by choosing t = {\alpha_3}{\alpha_6^{-1}}, we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle;

    4.\ if \alpha_7\neq0, then by choosing t = -{\alpha_2}{\alpha_7^{-1}}, we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5 \rangle,\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle,\langle \alpha\nabla_2+\beta\nabla_3+\nabla_4+\gamma\nabla_5 \rangle,\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{379}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{380}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ & & e_1e_4 = \beta e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{381}^{\alpha, \beta,\gamma,\mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 \\ && e_3e_3 = \mu e_5 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{382}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{09}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{09} & \begin{array}{l} e_1e_1 = e_2 \\ e_2e_2 = e_3 \\ e_3e_3 = e_4 \end{array} &\begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{09}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin\{ (1,1),(2,2),(3,3)\} \end{array} & \phi = \begin{pmatrix} x&0&0&0\\ 0&x^2&0&0\\ 0&0&x^4&0\\ t&0&0&x^8 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{09}) . Since

    \phi^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&\alpha_4&\alpha_5\\ \alpha_2&\alpha_4&0&\alpha_6\\ \alpha_3&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha^*_1&\alpha^{*}_2&\alpha^*_3\\ \alpha^*_1&0&\alpha^*_4&\alpha^*_5\\ \alpha^{*}_2&\alpha^*_4&0&\alpha^*_6\\ \alpha^*_3&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{llll} \alpha_1^* = (\alpha_1x+\alpha_5t)x^2, & \alpha_2^* = (\alpha_2x+\alpha_6t)x^4, & \alpha_3^* = (\alpha_3x+\alpha_7t)x^8, & \alpha_4^* = \alpha_4x^6, \\ \alpha_5^* = \alpha_5x^{10}, & \alpha_6^* = \alpha_6x^{12}, & \alpha_7^* = \alpha_7x^{16}. \end{array}

    Since (\alpha_3,\alpha_5,\alpha_6,\alpha_7)\neq(0,0,0,0), we have the following cases:

    1.\ \alpha_7 = \alpha_6 = \alpha_5 = 0, then \alpha_3\neq0, and we have the following subcases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_2 = 0, then we have representatives \langle \nabla_3 \rangle and \langle \nabla_3+\nabla_4 \rangle depending on whether \alpha_4 = 0 or not;

    {\rm{(b)}}\ if \alpha_1 = 0, \alpha_2\neq0, then by choosing x = \sqrt[4]{{\alpha_2}{\alpha_3^{-1}}}, t = 0, we have the family of representatives \langle \nabla_2+\nabla_3+\alpha\nabla_4 \rangle;

    {\rm{(c)}}\ if \alpha_1\neq0, then by choosing x = \sqrt[6]{{\alpha_1}{\alpha_3^{-1}}}, t = 0, we have the family of representatives \langle \nabla_1+\alpha\nabla_2+\nabla_3+\beta\nabla_4 \rangle.

    2. \alpha_7 = 0, \alpha_6 = 0, \alpha_5\neq0, then we have the following cases:

    {\rm{(a)}}\ if \alpha_2 = 0, \alpha_3 = 0, then we have representatives \langle \nabla_5 \rangle and \langle \nabla_4+\nabla_5 \rangle depending on whether \alpha_4 = 0 or not;

    {\rm{(b)}}\ if \alpha_2 = 0, \alpha_3\neq0, then choosing x = {\alpha_3}{\alpha_5^{-1}}, t = -{\alpha_1\alpha_3}{\alpha_5^{-2}}, we have the family of representatives \langle \nabla_3+ \alpha\nabla_4+\nabla_5 \rangle;

    {\rm{(c)}}\ if \alpha_2\neq0, then by choosing x = \sqrt[5]{{\alpha_2}{\alpha_5^{-1}}}, t = -{\alpha_1\sqrt[5]{\alpha_2 \alpha_5^{-6} }}, we have the family of representatives \langle \nabla_2+\alpha\nabla_3+ \beta\nabla_4+\nabla_5 \rangle.

    3. \alpha_7 = 0, \alpha_6\neq0, then we have the following cases:

    {\rm{(a)}}\ if \alpha_1\alpha_6 = \alpha_2\alpha_5, \alpha_3 = 0, \alpha_4 = 0, then we have representatives \langle \nabla_6 \rangle and \langle \nabla_5+\nabla_6 \rangle depending on whether \alpha_5 = 0 or not;

    {\rm{(b)}}\ if \alpha_1\alpha_6 = \alpha_2\alpha_5, \alpha_3 = 0, \alpha_4\neq0, then by choosing x = \sqrt[6]{{\alpha_4}{\alpha_6^{-1}}}, t = -{\alpha_2\sqrt[6]{\alpha_4 \alpha_6^{-7} }}, we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle;

    {\rm{(c)}}\ if \alpha_1\alpha_6 = \alpha_2\alpha_5, \alpha_3\neq0, then by choosing x = \sqrt[3]{{\alpha_3}{\alpha_6^{-1}}}, t = -{\alpha_2\sqrt[3]{\alpha_3 \alpha_6^{-4}}}, we have the family of representatives \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle;

    {\rm{(d)}}\ if \alpha_1\alpha_6\neq \alpha_2\alpha_5 , then by choosing

    x = \sqrt[9]{(\alpha_1\alpha_6-\alpha_2\alpha_5)\alpha^{-2}_6}, t = -{\alpha_2\sqrt[9]{(\alpha_1\alpha_6-\alpha_2\alpha_5)\alpha_6^{-11}}},

    we have the family of representatives

    \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle.

    4. \alpha_7\neq0, then we have the following cases:

    {\rm{(a)}}\ if \alpha_1 = 0, \alpha_2\alpha_7 = \alpha_3\alpha_6, \alpha_4 = 0, \alpha_5 = 0, then choosing x = 1, t = -\frac{\alpha_3}{\alpha_7}, we have representatives \langle \nabla_7 \rangle and \langle \nabla_6+\nabla_7 \rangle depending on whether \alpha_6 = 0 or not;

    {\rm{(b)}}\ if \alpha_1\alpha_7 = \alpha_3\alpha_5, \alpha_2\alpha_7 = \alpha_3\alpha_6, \alpha_4 = 0, \alpha_5\neq0, then by choosing x = \sqrt[6]{{\alpha_5}{\alpha_7^{-1}}}, t = -{\alpha_3\sqrt[6]{\alpha_5 \alpha_7^{-7} }}, we have the family of representatives \langle \nabla_5+\alpha\nabla_6+\nabla_7 \rangle;

    {\rm{(c)}}\ if \alpha_1\alpha_7 = \alpha_3\alpha_5, \alpha_2\alpha_7 = \alpha_3\alpha_6, \alpha_4\neq0, then by choosing

    x = \sqrt[10]{{\alpha_4}{\alpha_7^{-1}}}, t = -{\alpha_3\sqrt[10]{\alpha_4 \alpha_7^{-11}}},

    we have the family of representatives \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle;

    {\rm{(d)}}\ \alpha_1\alpha_7 = \alpha_3\alpha_5, \alpha_2\alpha_7\neq\alpha_3\alpha_6, then by choosing

    x = \sqrt[11]{(\alpha_2\alpha_7-\alpha_3\alpha_6)\alpha^{-2}_7 }, t = -{\alpha_3\sqrt[11]{(\alpha_2\alpha_7-\alpha_3\alpha_6) \alpha_7^{-13} }},

    we have the family of representatives

    \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle;

    {\rm{(e)}}\ \alpha_1\alpha_7\neq\alpha_3\alpha_5, then by choosing

    x = \sqrt[13]{({\alpha_1\alpha_7-\alpha_3\alpha_5}){\alpha^{-2}_7}}, t = -{\alpha_3\sqrt[13]{(\alpha_1\alpha_7-\alpha_3\alpha_5)\alpha_7^{-15} }},

    we have the family of representatives

    \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \nabla_1+\alpha\nabla_2+\nabla_3+\beta\nabla_4 \rangle ^{O(\alpha,\beta) = O(-\eta_3\alpha,\beta) = O(-\eta_3\alpha,-\beta) = O(\eta_3^2\alpha,-\beta) = O(\eta^2_3\alpha,\beta) = O(\alpha,-\beta)}, \\ \langle \nabla_1+\alpha\nabla_2+\beta\nabla_4+\gamma\nabla_5+\mu\nabla_6+ \\ \nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\eta_{13}^{11}\alpha,\eta^{10}_{13}\beta,\eta^6_{13}\gamma,\eta^4_{13}\mu) = \\ O(-\eta^9_{13}\alpha,-\eta^7_{13}\beta,\eta^{12}_{13}\gamma,\eta^8_{13}\mu) = O(-\eta^7_{13}\alpha,\eta^4_{13}\beta,-\eta^5_{13}\gamma,\eta^{12}_{13}\mu) = \\ O(-\eta^5_{13}\alpha,-\eta^{1}_{13}\beta,-\eta^{11}_{13}\gamma,-\eta^{3}_{13}\mu) = O(-\eta^{3}_{13}\alpha,-\eta^{11}_{13}\beta,\eta^4_{13}\gamma,-\eta^7_{13}\mu) = \\ O(-\eta^{1}_{13}\alpha,\eta^8_{13}\beta,\eta^{10}_{13}\gamma,-\eta^{11}_{13}\mu) = O(\eta^{12}_{13}\alpha,-\eta^5_{13}\beta,-\eta^{3}_{13}\gamma,\eta^{2}_{13}\mu) = \\ O(\eta^{10}_{13}\alpha,\eta^{2}_{13}\beta,-\eta^{9}_{13}\gamma,\eta^6_{13}\mu) = O(\eta^8_{13}\alpha,\eta^{12}_{13}\beta,\eta^{2}_{13}\gamma,\eta^{10}_{13}\mu) = \\ O(\eta^6_{13}\alpha,-\eta^{9}_{13}\beta,\eta^{8}_{13}\gamma,-\eta^{1}_{13}\mu) = O(\eta^4_{13}\alpha,\eta^{6}_{13}\beta,-\eta^{1}_{13}\gamma,-\eta^5_{13}\mu) = \\ O(\eta^{2}_{13}\alpha,-\eta^{3}_{13}\beta,-\eta^{7}_{13}\gamma,-\eta^9_{13}\mu) \end{array}}}, \\ \langle \nabla_1+\alpha\nabla_3+\beta\nabla_4+\gamma\nabla_5+\nabla_6 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma,\mu) = O(-\eta_3\alpha,\eta^2_3\beta,\eta^2_9\gamma) = O(\eta^2_3\alpha,-\eta_3\beta,\eta^4_9\gamma) = \\ O(\alpha,\beta,\eta^2_3\gamma) = O(-\eta_3\alpha,\eta^2_3\beta,\eta^8_9\gamma) = O(\eta^2_3\alpha,-\eta_3\beta,-\eta_9\gamma) = \\ O(\alpha,\beta,-\eta_3\gamma) = O(-\eta_3\alpha,\eta^2_3\beta,-\eta^5_9\gamma) = O(\eta^2_3\alpha,-\eta_3\beta,-\eta^7_9\gamma) \end{array}}}, \\ \langle \nabla_2+\nabla_3+\alpha\nabla_4 \rangle^{O(\alpha) = O(i\alpha) = O(-\alpha) = (-i\alpha)}, \\ \langle \nabla_2+\alpha\nabla_3+ \beta\nabla_4+\nabla_5 \rangle^{ O(\alpha,\beta) = O(-\eta_5\alpha,\eta^4_5\beta) = O(\eta^2_5\alpha,-\eta^3_5\beta) = O(-\eta^3_5\alpha,\eta^2_5\beta) = O(\eta_5^4\alpha,-\eta_5\beta)},\\ \langle \nabla_2+\alpha\nabla_4+\beta\nabla_5+\gamma\nabla_6+\nabla_7 \rangle^{{\begin{array}{l} O(\alpha,\beta,\gamma) = O(\eta_{11}^{10}\alpha,\eta^6_{11}\beta,\eta^4_{11}\gamma = \\ O(-\eta^9_{11}\alpha,-\eta_{11}\beta,\eta^38_{11}\gamma) = O(\eta^8_{11}\alpha,-\eta^7_{11}\beta,-\eta_{11}\gamma) = \\ O(-\eta^7_{11}\alpha,\eta^2_{11}\beta,-\eta^5_{11}\gamma) = O(\eta^{6}_{11}\alpha,\eta^8_{11}\beta,-\eta^9_{11}\gamma) = \\ O(-\eta^{5}_{11}\alpha,-\eta^3_{11}\beta,\eta^2_{11}\gamma) = O(\eta^4_{11}\alpha,-\eta^9_{13}\beta,\eta^{6}_{11}\gamma) = \\ O(-\eta^3_{11}\alpha,\eta^{4}_{11}\beta,\eta^{10}_{11}\gamma) = O(\eta^2_{11}\alpha,\eta^{10}_{11}\beta,-\eta^{3}_{11}\gamma) = \\ O(-\eta_{11}\alpha,-\eta^{5}_{11}\beta,-\eta^{7}_{11}\gamma) \end{array}}}, \langle \nabla_3 \rangle, \\ \langle \nabla_3+\nabla_4 \rangle, \langle \nabla_3+ \alpha\nabla_4+\nabla_5 \rangle, \langle \nabla_3+\alpha\nabla_4+\beta\nabla_5+\nabla_6 \rangle^{O(\alpha,\beta) = O(\alpha,-\eta_3\beta) = O(\alpha,\eta^2_3\beta)}, \\ \langle \nabla_4+\nabla_5 \rangle \langle \nabla_4+\alpha\nabla_5+\nabla_6 \rangle^{{ \begin{array}{l}O(\alpha) = O(-\eta_3\alpha) = \\O(\eta^2_3\alpha) \end{array}}}, \\ \langle \nabla_4+\alpha\nabla_5+\beta\nabla_6+\nabla_7 \rangle^{ { \begin{array}{l} O(\alpha,\beta) = O(-\eta_5\alpha,\eta^4_5\beta) = O(\eta^2_5\alpha,-\eta^3_5\beta) = \\ O(-\eta^3_5\alpha,\eta^2_5\beta) = O(\eta^4_5\alpha,-\eta_5 \beta) \end{array}}}, \langle \nabla_5 \rangle, \langle \nabla_5+\nabla_6 \rangle, \\ \langle \nabla_5+\alpha\nabla_6+\nabla_7 \rangle^{O(\alpha) = O(-\eta_3\alpha) = O(\eta^2_3\alpha)}, \langle \nabla_6 \rangle, \langle \nabla_6+\nabla_7 \rangle, \langle \nabla_7 \rangle,\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{383}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{384}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{385}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{386}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{387}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{388}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{389} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_3e_3 = e_4 \\ {\mathbf{N}}_{390} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{391}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{392}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{393} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{394}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{395}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{396} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{397} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{398}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{399} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{400} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{401} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{10}^{4}:

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{10} & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = e_4 \\ e_2e_2 = e_3 \\ e_3e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{10}) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(1,2),(2,2)\} \end{array} & \begin{array}{l} \phi_k = \begin{pmatrix} \eta^k&0&0&0\\ 0& \eta^{2k}&0&0\\ 0&0& \eta^{4k}&0\\ t&0&0& \eta^{8k} \end{pmatrix} \\ {\eta = -\eta_5, \ k = 0,1,2,3,4} \end{array}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{llll} \nabla_1 = [\Delta_{13}], & \nabla_2 = [\Delta_{14}], & \nabla_3 = [\Delta_{23}], & \nabla_4 = [\Delta_{24}], \\ \nabla_5 = [\Delta_{33}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{10}) . Since

    \phi_k^T\begin{pmatrix} 0&0&\alpha_1&\alpha_2\\ 0&0&\alpha_3&\alpha_4\\ \alpha_1&\alpha_3&\alpha_5&\alpha_6\\ \alpha_2&\alpha_4&\alpha_6&\alpha_7 \end{pmatrix}\phi_k = \begin{pmatrix} \alpha^*&\alpha^{**}&\alpha^{*}_1&\alpha^*_2\\ \alpha^{**}&0&\alpha^*_3&\alpha^*_4\\ \alpha^{*}_1&\alpha^*_3&\alpha^*_5+\alpha^{**}&\alpha^*_6\\ \alpha^*_2&\alpha^*_4&\alpha^*_6&\alpha^*_7 \end{pmatrix},

    we have

    \begin{array}{llll} \alpha_1^* = \eta^{4 k} (\eta^k \alpha_1+t \alpha_6), & \alpha_2^* = \eta^{8 k} (\eta^k \alpha_2+t \alpha_7), & \alpha_3^* = \eta^{6 k} \alpha_3, & \alpha_4^* = \eta^{10 k} \alpha_4, \\ \alpha_5^* = -t \eta^{2 k} \alpha_4+\eta^{8 k} \alpha_5, & \alpha_6^* = \eta^{12 k} \alpha_6, & \alpha_7^* = \eta^{16 k} \alpha_7. \end{array}

    Since (\alpha_2,\alpha_4,\alpha_6,\alpha_7)\neq(0,0,0,0), we have the following cases:

    1.\ if \alpha_7 = 0, \alpha_6 = 0, \alpha_4 = 0, then \alpha_2\neq0, and we have the family of representatives

    \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5\rangle;

    2.\ if \alpha_7 = 0, \alpha_6 = 0, \alpha_4\neq0, then we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\nabla_4 \rangle;

    3.\ if \alpha_7 = 0, \alpha_6\neq0, then we have the family of representatives

    \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle;

    4.\ if \alpha_7\neq0, then we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_3+\nabla_4 \rangle^{ {\begin{array}{l} O(\alpha, \beta, \gamma) = O(\alpha, \eta^4_5\beta, -\eta_5 \gamma) = O(\alpha, -\eta^3_5\beta, \eta^2_5 \gamma) = \\ O(\alpha, \eta^2_5\beta, -\eta^3_5 \gamma) = O(\alpha, -\eta_5\beta, \eta^4_5 \gamma)\end{array}}}, \\ \langle \alpha\nabla_1+\nabla_2+\beta\nabla_3+\gamma\nabla_5 \rangle^{ {\begin{array}{l} O(\alpha, \beta, \gamma) = O(-\eta_5\alpha, \eta^2_5\beta, \eta^4_5 \gamma) = O(\eta^2_5\alpha, \eta^4_5\beta, -\eta^3_5 \gamma) = \\ O(-\eta^3_5\alpha, -\eta_5\beta, \eta^2_5 \gamma) = O(\eta^4_5\alpha, -\eta^3_5\beta, -\eta_5 \gamma) \end{array}}}, \\ \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+ \\ \nabla_7 \rangle^{{\begin{array}{l} O(\alpha, \beta, \gamma, \mu, \nu) = O(\eta^4_5\alpha, \beta, \eta^4_5 \gamma, \eta^2_5\mu, -\eta_5\nu) = \\ O(-\eta^3_5\alpha, \beta, -\eta^3_5 \gamma, \eta^4_5\mu, \eta^2_5\nu) = O(\eta^2_5\alpha, \beta, \eta^2_5 \gamma,- \eta_5\mu, -\eta^3_5\nu) = \\ O(-\eta_5\alpha, \beta, -\eta_5 \gamma, -\eta^3_5\mu, \eta^4_5\nu) \end{array}}}, \\ \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+ \\ \nabla_6 \rangle^{{\begin{array}{l} O(\alpha, \beta, \gamma, \mu) = O(\eta_5^2\alpha, \eta_5^4\beta, -\eta_5^3 \gamma, -\eta_5\mu) = O(\eta_5^4\alpha, -\eta_5^3\beta, -\eta_5\gamma, \eta^2_5\mu) = \\ O(-\eta_5\alpha, \eta_5^2\beta, \eta_5^4 \gamma, -\eta^3_5\mu) = O(-\eta_5^3\alpha, -\eta_5\beta, \eta_5^2 \gamma, \eta^4_5\mu) \end{array}}},\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{402}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{403}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4+\gamma e_5 \\ {\mathbf{N}}_{404}^{\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ & & e_3e_3 = e_4+\mu e_5 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{405}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_3e_4 = e_5 \\ \end{array}

    Here we will collect all information about {\mathbf N}_{11}^{4}(\lambda):

    \begin{array}{|l|l|l|l|} \hline \rm{ } & \rm{ } & \rm{Cohomology} & \rm{Automorphisms} \\ \hline {\mathbf{N}}^{4}_{11}(\lambda) & \begin{array}{l} e_1e_1 = e_2 \\ e_1e_2 = \lambda e_4 \\ e_2e_2 = e_3\\ e_2e_3 = e_4 \\ e_3e_3 = e_4 \end{array} & \begin{array}{l} \mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{11}(\lambda)) = \Big \langle [\Delta_{ij}] \Big\rangle\\ (i,j) \notin \{ (1,1),(2,2),(3,3) \} \end{array} & \phi = \begin{pmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ t&0&0&1 \end{pmatrix}\\ \hline \end{array}

    Let us use the following notations:

    \begin{array}{lll l} \nabla_1 = [\Delta_{12}], & \nabla_2 = [\Delta_{13}], & \nabla_3 = [\Delta_{14}], & \nabla_4 = [\Delta_{23}], \\ \nabla_5 = [\Delta_{24}], & \nabla_6 = [\Delta_{34}], & \nabla_7 = [\Delta_{44}]. \end{array}

    Take \theta = \sum\limits_{i = 1}^{7}\alpha_i\nabla_i\in\mathrm{H}^2_{\mathfrak{C}}(\mathbf{N}^{4}_{11}(\lambda)) . Since

    \phi^T\begin{pmatrix} 0&\alpha_1&\alpha_2&\alpha_3\\ \alpha_1&0&\alpha_4&\alpha_5\\ \alpha_2&\alpha_4&0&\alpha_6\\ \alpha_3&\alpha_5&\alpha_6&\alpha_7 \end{pmatrix}\phi = \begin{pmatrix} \alpha^*&\alpha_1^{*}&\alpha^{*}_2&\alpha^*_3\\ \alpha_1^{*}&0&\alpha^*_4&\alpha^*_5\\ \alpha^{*}_2&\alpha^*_4&0&\alpha^*_6\\ \alpha^*_3&\alpha^*_5&\alpha^*_6&\alpha^*_7 \end{pmatrix}

    we have

    \begin{array}{llll} \alpha_1^* = \alpha_1+\alpha_5t, & \alpha_2^* = \alpha_2+\alpha_6t, & \alpha_3^* = \alpha_3+\alpha_7t, & \alpha_4^* = \alpha_4, \\ \alpha_5^* = \alpha_5, & \alpha_6^* = \alpha_6, & \alpha_7^* = \alpha_7. \end{array}

    Since (\alpha_3,\alpha_5,\alpha_6,\alpha_7)\neq(0,0,0,0), we have the following cases:

    1.\ if \alpha_7 = 0, \alpha_6 = 0, \alpha_5 = 0 , then \alpha_3\neq0 and we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_2+\nabla_3+\gamma\nabla_4 \rangle;

    2.\ if \alpha_7 = 0, \alpha_6 = 0, \alpha_5\neq0 then by choosing t = -{\alpha_1}{\alpha_5^{-1}}, we have the family of representatives

    \langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\nabla_5 \rangle;

    3.\ if \alpha_7 = 0, \alpha_6\neq0 then by choosing t = -{\alpha_2}{\alpha_6^{-1}}, we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle;

    4.\ if \alpha_7\neq0 then by choosing t = -{\alpha_3}{\alpha_7^{-1}}, we have the family of representatives

    \langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle.

    Summarizing, we have the following distinct orbits:

    \begin{array}{c} \langle \alpha\nabla_1+\beta\nabla_2+\nabla_3+\gamma\nabla_4 \rangle,\langle \alpha\nabla_1+\beta\nabla_2+\gamma\nabla_4+\mu\nabla_5+\nu\nabla_6+\nabla_7 \rangle, \\\langle \alpha\nabla_1+\beta\nabla_3+\gamma\nabla_4+\mu\nabla_5+\nabla_6 \rangle,\langle \alpha\nabla_2+\beta\nabla_3+\gamma\nabla_4+\nabla_5 \rangle,\end{array}

    which gives the following new algebras:

    \begin{array}{llllllllllllllllll} {\mathbf{N}}_{406}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4+\alpha e_5 & e_1e_3 = \beta e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{407}^{\lambda,\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4+\alpha e_5 & e_1e_3 = \beta e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 & e_2e_4 = \mu e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{408}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4+\alpha e_5 & e_1e_4 = \beta e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\gamma e_5 & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{409}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \end{array}

    Remark 2. Note that the algebras \mathbf{N}^{4}_{11}(\lambda) and \mathbf{N}^{4}_{11}(-\lambda) are isomorphic. Hence, there are some additional isomorphism relations for algebras from the present subsection \mathbf{N}^{\lambda, \Xi} \cong \mathbf{N}^{-\lambda, \Xi} .

    Theorem 5.1. Let \mathbf N be a complex 5 -dimensional nilpotent commutative algebra. Then we have one of the following situations.

    {\mathit{1.}} \ If \mathbf N is associative, then \mathbf N is isomorphic to one algebra listed in [15].

    {\mathit{2.}} \ If \mathbf N is a non-associative Jordan algebra, then \mathbf N is isomorphic to one algebra listed in [9].

    {\mathit{3.}} \ If \mathbf N is a non-Jordan \mathfrak{CD} -algebra, then \mathbf N is isomorphic to one algebra listed in [11].

    {\mathit{4.}}\ If \mathbf N is a non- \mathfrak{CD} -algebra, then \mathbf N is isomorphic to one algebra listed in the following list.

    \begin{array}{llllllll} {\mathbf N}_{01} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_2e_3 = e_4 \\ {\mathbf N}_{02} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_1e_3 = e_4 & e_2 e_3 = e_4 && \\ {\mathbf N}_{03} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_3 e_3 = e_4 &&\\ {\mathbf N}_{04} & : & e_1 e_1 = e_2 & e_1 e_2 = e_3 & e_2e_2 = e_4 & e_3 e_3 = e_4 &&\\ {\mathbf N}_{05} & : & e_1 e_1 = e_2 & e_1 e_3 = e_4 & e_2 e_2 = e_3 && \\ {\mathbf N}_{06} & : & e_1 e_1 = e_2 & e_1e_2 = e_4 & e_1 e_3 = e_4 & e_2 e_2 = e_3 && \\ {\mathbf N}_{07} & : & e_1 e_1 = e_2 & e_2 e_2 = e_3 & e_2 e_3 = e_4 && \\ {\mathbf N}_{08} & : & e_1 e_1 = e_2 & e_1e_3 = e_4 & e_2 e_2 = e_3 & e_2 e_3 = e_4 && \\ {\mathbf N}_{09} & : & e_1 e_1 = e_2 & e_2 e_2 = e_3 & e_3 e_3 = e_4 && \\ {\mathbf N}_{10} & : & e_1 e_1 = e_2 & e_2e_2 = e_3 & e_1e_2 = e_4 & e_3 e_3 = e_4 &&\\ {\mathbf N}_{11}^{\lambda} & : & e_1 e_1 = e_2 & e_1e_2 = \lambda e_4 & e_2 e_2 = e_3 \\ && e_2e_3 = e_4 & e_3 e_3 = e_4 &\\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{12} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{13}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_4 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{14} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{15} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{16} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{17} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{18} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{19} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{20}^{\alpha} & : & e_1e_1 = e_2 & { e_1e_2 = e_4+\alpha e_5 }& e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{21} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{22}^{\alpha\neq 1} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{23}^{\alpha, \beta} & : & e_1e_1 = e_2 & { e_1e_2 = \beta e_4 +\alpha e_5 }& e_1e_3 = e_4 \\ && e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{24}^{\alpha} & : & e_1e_1 = e_2 & { e_1e_2 = \alpha e_4+e_5 }& e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{25} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{26} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_2e_2 = e_3 & e_3e_3 = e_4 \\ {\mathbf{N}}_{27} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{28} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{29} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{30} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{31} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{32}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{33} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{34} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{35} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{36} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{37} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{38}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{39} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{40} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{41} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{42} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{43} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{44} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{45} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{46} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{47} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_5 \\ && e_2e_2 = e_3 & e_4e_4 = e_5 \\ {\mathbf{N}}_{48} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{49}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{50}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_2e_2 = e_3 & e_2e_3 = \beta e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{51}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{52} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{53}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{54} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{55} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{56} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{57} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{58} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{59} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ {\mathbf{N}}_{60} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 \\ & & e_2e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{61} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_4e_4 = e_5 \\ {\mathbf{N}}_{62} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ {\mathbf{N}}_{63}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{64} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{65} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_3e_3 = e_5 \\ {\mathbf{N}}_{66} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{67} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{68} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{69} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{70} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{71} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{72} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_1e_4 = \frac{3}{4}e_5 & e_2e_2 = e_4 & e_2e_3 = -\frac{3}{4}e_5 \\ {\mathbf{N}}_{73}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = 3e_5 & e_2e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{74}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_1e_4 = \alpha e_5 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{75}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_2e_2 = e_4 & e_2e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{76}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & { e_1e_4 = (1+\alpha) e_5 } \\ && e_2e_2 = e_4 & e_2e_3 = 3\alpha e_5 \\ {\mathbf{N}}_{77}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = 3e_5 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{78}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = 3\alpha e_5 & e_2e_4 = \beta e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{79} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{80}^{\alpha \neq 1} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_4 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{81}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{82} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{83}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{84} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 & e_2e_3 = e_4 \\ {\mathbf{N}}_{85}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{86}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{87} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{88}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{89}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{90}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{91} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{92} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{93} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{94} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{95}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{96}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{97} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{98} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{99}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{100} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{101} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{102} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 \\ {\mathbf{N}}_{103} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ {\mathbf{N}}_{104}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{105}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{106} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{107} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{108} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{109} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{110} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{111} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{112}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{113} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{114} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{115} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{116} & : & e_1e_1 = e_2 & e_2e_2 = e_5 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{117} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{118} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{119} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{120} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{121} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_2e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{122} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{123} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{124} & : & e_1e_1 = e_2 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}^{\alpha}_{125} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}^{\alpha,\beta}_{126} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_2e_2 = e_5 \\ & & e_3e_3 = e_4 + \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}^{\alpha}_{127} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{128} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = e_5 \\ && e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{129} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{130} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_2 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{131} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{132} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4+e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{133} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{134} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{135}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{136}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 \\ && e_2e_2 = e_4+\alpha e_5 & e_2e_3 = \beta e_5 \\ && e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{137}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{138}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = 4e_5 \\ {\mathbf{N}}_{139}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{140}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{141}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{142}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4+e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{143}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{144}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{145}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{146} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{147} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{148} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{149} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{150}^{\alpha, \beta, \gamma, \mu } & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{151}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4+\beta e_5 \\ {\mathbf{N}}_{152}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{153}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & { e_3e_3 = e_4+\beta e_5 } & e_3e_4 = \gamma e_5 \\ {\mathbf{N}}_{154}^{\alpha, \beta, \gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & { e_3e_3 = e_4+\gamma e_5 } & e_4e_4 = e_5 \\ {\mathbf{N}}_{155} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_3e_3 = e_4 \\ {\mathbf{N}}_{156}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_2e_3 = e_5 \\ & & { e_3e_3 = e_4+\alpha e_5 } \\ {\mathbf{N}}_{157}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{158}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4+\alpha e_5 & e_3e_4 = \beta e_5 \\ {\mathbf{N}}_{159} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_4 \\ & & e_3e_3 = e_4+e_5 \\ {\mathbf{N}}_{160}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4+\beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{161} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_3 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{162}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = e_4+e_5 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{163}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4+\alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{164}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 \\ {\mathbf{N}}_{165} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4+e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{166} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{167} & : & e_1e_1 = e_2 & e_2e_2 = e_4 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{168}^{\lambda \neq 1; 2} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_1e_4 = (\lambda-4)e_5 & { e_2e_2 = \lambda e_4 + 4(1-\lambda)(\lambda-2)e_5 } \\ && { e_2e_3 = - \lambda(\lambda+2)e_5 }\\ {\mathbf{N}}_{169}^{\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = -2 e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{170}^{\lambda, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_2e_2 = \lambda e_4 & { e_2e_3 = (1+\alpha(3\lambda-2)) e_5 } \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{171}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_1e_4 = e_5 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = (3\lambda-2) e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{172}^{\lambda \neq 0,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 + e_5 & e_2e_3 = \alpha e_5 & e_2e_4 = \frac{\lambda} {4} e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{173}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{174}^{\lambda\neq-2, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 + \alpha e_5 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{175}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_2e_2 = -2 e_4+\alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{176}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4+\alpha e_5 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{177}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4+ e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{178}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_2e_2 = \lambda e_4 & e_2e_3 = e_5 & e_2e_4 = e_5 \\ {\mathbf{N}}_{179}^{\lambda,\alpha\neq0} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{180}^{\lambda,\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \lambda e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{181}^{\alpha \neq 0 } & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = -2 e_4 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{182} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = -2 e_4 & e_2e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{183}^{\lambda\neq 2} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{184}^{\lambda, \alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_2e_4 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{185}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{186}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{187}^{\lambda} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = \lambda e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{188}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{189}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{190} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\\end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{191}^{\alpha} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{192} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{193} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{194} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{195} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{196} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ {\mathbf{N}}_{197} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ {\mathbf{N}}_{198} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{199}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{200} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{201} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{202}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{203} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{204} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{205} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{206}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{207}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{208} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{209}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_5 \\ && e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{210}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{211} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{212} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{213} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{214} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{215}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{216}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = e_4+\alpha e_5 & e_2e_3 = \beta e_5 \\ & & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{217}^{\alpha} & : & e_1e_1 = \alpha e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{218}^\alpha & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ && e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{219} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = 2e_5 \\ && e_2e_2 = e_4+e_5 & e_2e_3 = e_5 \\ {\mathbf{N}}_{220}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4+\alpha e_5 & e_3e_3 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{221}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{222}^{\alpha\neq 0} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{223}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{224}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_4 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{225}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4+e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{226}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ & & e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{227}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{228} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{229} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{230} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{231} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{232}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{233}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{234}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4+e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{235}^{\alpha, \beta, \gamma} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{236}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{237}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4+e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{238}^{\alpha\neq0, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{239}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{240}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4+e_5 \\ & & e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{241} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{242}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ & & e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{243} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{244} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{245} & : & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{246}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{247}^{\alpha, \beta} & : & e_1e_1 = e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{248} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{249}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{250} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{251}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{252}^{\alpha, \beta} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{253}^{\alpha} & : & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{254} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{255}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{256}^{\alpha} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{257} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{258} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{259} & : & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{260}^{\alpha} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{261} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{262} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{263} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{264} & : & e_1e_2 = e_3 & e_2e_2 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{265} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{266} & : & e_1e_2 = e_3 & e_2e_3 = e_5 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{267} & : & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{268} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{269} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{270} & : & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{271}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_2e_2 = \beta e_5 \\ && e_2e_3 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{272}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{273}^{\alpha, \beta} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{274}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4+e_5 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = \beta e_5 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{275}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{276}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{277}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{278} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{279}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{280}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{281} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{282}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{283}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{284}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{285}^{\alpha} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{286} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{287} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{288}^{\alpha, \beta, \gamma,\mu,\nu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_3 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_3 = \mu e_5 & e_3e_3 = e_4 \\ & & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{289}^{\alpha, \beta, \gamma,\mu} & : & e_1e_1 = e_4+\alpha e_5 & e_1e_2 = e_3 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_4 = \mu e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{290}^{\alpha, \beta} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{291}^{\alpha, \beta\neq0 ,\gamma} & : & e_1e_1 = e_4 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{292}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ & & e_1e_4 = \alpha e_5 & e_2e_2 = \beta e_5 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = -\alpha e_5 \\ {\mathbf{N}}_{293}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{294}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{295}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 & e_2e_2 = e_5 \\ && e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{296}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_2 = e_5 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{297}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{298}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{299}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{300}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_4 & e_3e_3 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{301} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ {\mathbf{N}}_{302} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{303}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{304}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{305} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{306} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{307} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{308} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{309}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 \\ & & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4+\beta e_5 & e_3e_3 = -e_5 \\ {\mathbf{N}}_{310}^{\alpha, \beta\neq-1} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{311}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{312}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{313}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_4 & e_2e_2 = \alpha e_5 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{314}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{315}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = \beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{316}^{\alpha, \beta, \gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = e_5 \\ && e_2e_2 = \alpha e_5 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{317} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{318}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ && e_2e_2 = e_5 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{319}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{320}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 & e_2e_2 = \alpha e_5 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{321} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = e_5 \\ & & e_2e_3 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{322}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{323}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{324}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_2 = e_5 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{325} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{326}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{327}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{328} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{329} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}^\alpha_{330} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{331} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{332} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{333} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{334}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{335}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ && e_2e_2 = e_4+\gamma e_5 & e_2e_3 = \mu e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{336}^{\alpha, \beta,\gamma, \mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{337}^{\alpha, \beta,\gamma, \mu} & : & e_1e_1 = e_2 & e_1e_2 = e_3 & e_1e_4 = \alpha e_5 \\ && e_2e_2 = e_4+\beta e_5 & e_2e_3 = \gamma e_5 & e_2e_4 = \mu e_5 \\ && e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{338}^{\alpha} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = -2e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{339}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{340}^{\alpha, \beta ,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{341}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ {\mathbf{N}}_{342}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{343}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{344}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{345}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{346}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{347}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{348} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ {\mathbf{N}}_{349} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_2e_4 = e_5 & e_3e_3 = e_5 \\ {\mathbf{N}}_{350} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{351}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{352} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ && e_3e_3 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{353} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_3e_4 = e_5 \\ {\mathbf{N}}_{354} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_4e_4 = e_5 \\ {\mathbf{N}}_{355}^{\alpha\neq0, \beta} & : & e_1e_1 = e_2 & { e_1e_3 = e_4+\alpha e_5 } \\ && e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & { e_3e_3 = (\beta+2)e_5 } \\ {\mathbf{N}}_{356}^{\alpha\neq0, \beta,\gamma} & : & e_1e_1 = e_2 & { e_1e_3 = e_4+\alpha e_5 } \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = e_5 \\ & & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{357}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & { e_1e_3 = e_4+\alpha e_5 } \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = \gamma e_5 \\ & & e_3e_4 = \mu e_5 & e_4e_4 = e_5 & \\ {\mathbf{N}}_{358}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{359}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 & \\ {\mathbf{N}}_{360}^{\alpha,\beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ & & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{361}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \beta e_5 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{362}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = \beta e_5 \\ & & e_3e_3 = \gamma e_5 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{363}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = \beta e_5 & e_3e_3 = \gamma e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{364}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 \ & e_3e_3 = \beta e_5 \\ {\mathbf{N}}_{365}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{366}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_1e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{367} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ {\mathbf{N}}_{368}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4 & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{369}^{\alpha,\beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_2e_4 = \alpha e_5 & e_3e_3 = \beta e_5 & e_3e_4 = e_5 & \\ {\mathbf{N}}_{370} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4 & e_3e_3 = e_5 \\ {\mathbf{N}}_{371}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_2e_4 = e_5 & e_3e_3 = \alpha e_5 \\ {\mathbf{N}}_{372}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = e_5 & e_3e_3 = \alpha e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{373}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_2e_4 = e_5 \\ && e_3e_3 = \alpha e_5 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{374} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{375}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ && e_3e_3 = e_5 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{376} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{377} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{378} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{379}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ && e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4+\beta e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{380}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 \\ & & e_1e_4 = \beta e_5 & e_2e_2 = e_3 & e_2e_3 = e_4 \\ & & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{381}^{\alpha, \beta,\gamma,\mu, \nu} & : & e_1e_1 = e_2 & e_1e_2 = \alpha e_5 & e_1e_3 = e_4 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = \mu e_5 \\ & & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{382}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = e_4+\beta e_5 & e_2e_4 = e_5 & e_3e_3 = \gamma e_5 \\ {\mathbf{N}}_{383}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{384}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \mu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{385}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{386}^{\alpha} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = e_5 \\ && e_2e_2 = e_3 & e_2e_3 = \alpha e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{387}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \beta e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{388}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_3 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = \gamma e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{389} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_3e_3 = e_4 \\ {\mathbf{N}}_{390} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_5 & e_3e_3 = e_4 \\ \end{array}
    \begin{array}{llllllll} {\mathbf{N}}_{391}^{\alpha} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = \alpha e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{392}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_1e_4 = e_5 & e_2e_2 = e_3 & e_2e_3 = \alpha e_5 \\ && e_2e_4 = \beta e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{393} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ && e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{394}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 \\ & & e_2e_4 = \alpha e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{395}^{\alpha, \beta} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_3 = e_5 & e_2e_4 = \alpha e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = \beta e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{396} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{397} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ & & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{398}^{\alpha} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_2e_4 = e_5 \\ && e_3e_3 = e_4 & e_3e_4 = \alpha e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{399} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{400} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 \\ && e_3e_4 = e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{401} & : & e_1e_1 = e_2 & e_2e_2 = e_3 & e_3e_3 = e_4 & e_4e_4 = e_5 \\ {\mathbf{N}}_{402}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = \beta e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \gamma e_5 & e_2e_4 = e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{403}^{\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 & e_1e_4 = e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_3e_3 = e_4+\gamma e_5 \\ {\mathbf{N}}_{404}^{\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_3 = \alpha e_5 \\ & & e_2e_2 = e_3 & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 \\ & & e_3e_3 = e_4+\mu e_5 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{405}^{\alpha, \beta,\gamma,\mu} & : & e_1e_1 = e_2 & e_1e_2 = e_4 & e_1e_4 = \alpha e_5 & e_2e_2 = e_3 \\ & & e_2e_3 = \beta e_5 & e_2e_4 = \gamma e_5 & e_3e_3 = e_4+\mu e_5 & e_3e_4 = e_5 \\ {\mathbf{N}}_{406}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & { e_1e_2 = \lambda e_4+\alpha e_5 } \\ && e_1e_3 = \beta e_5 & e_1e_4 = e_5 & e_2e_2 = e_3 \\ && e_2e_3 = e_4+\gamma e_5 & e_3e_3 = e_4 \\ {\mathbf{N}}_{407}^{\lambda,\alpha, \beta,\gamma,\mu,\nu} & : & e_1e_1 = e_2 & { e_1e_2 = \lambda e_4+\alpha e_5 } \\ && e_1e_3 = \beta e_5 & e_2e_2 = e_3 & { e_2e_3 = e_4+\gamma e_5 } \\ & & e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = \nu e_5 & e_4e_4 = e_5 \\ {\mathbf{N}}_{408}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & { e_1e_2 = \lambda e_4+\alpha e_5 } \\ & & e_1e_4 = \beta e_5 & e_2e_2 = e_3 & e_2e_3 = e_4+\gamma e_5 \\ && e_2e_4 = \mu e_5 & e_3e_3 = e_4 & e_3e_4 = e_5 \\ {\mathbf{N}}_{409}^{\lambda,\alpha, \beta,\gamma} & : & e_1e_1 = e_2 & e_1e_2 = \lambda e_4 & e_1e_3 = \alpha e_5 \\ && e_1e_4 = \beta e_5 & e_2e_2 = e_3 & { e_2e_3 = e_4+\gamma e_5 } \\ & & e_2e_4 = e_5 & e_3e_3 = e_4 \end{array}
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