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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Representations of Six-dimensional Mubarakazyanov Lie algebras

G Thompson1*, C Hettiarachchi2, N Jones3 and A Shabanskaya4

1Department of Mathematics & Statistics, The University of Toledo, 2801 W. Bancroft St., Toledo, OH 43606, USA

2Department of Mathematics, Northern Virginia Community College, USA

3Department of Mathematics & Statistics, The University of Toledo, 2801 W. Bancroft St., Toledo, OH 43606, USA

4Department of Mathematics and Computer Science, Eastern Connecticut State University, 83 Windham Street, Willimantic, CT 06226, USA

*Corresponding Author:
Thompson
Department of Mathematics and Statistics
The University of Toledo, 2801 W. Bancroft St.
Toledo, OH 43606, USA
Tel: +7348379283
E-mail: [email protected]

Received date: September 05, 2013; Accepted date: May 02, 2014; Published date: May 10, 2014

Citation: Thompson G, Hettiarachchi C, Jones N, Shabanskaya A (2014) Representations of Six-dimensional Mubarakazyanov Lie algebras. J Generalized Lie Theory Appl 8:211. doi:10.4172/1736-4337.1000211

Copyright: © 2014 Thompson G. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

A corrected list of the Mubarakzyanov algebras, that is, six-dimensional indecomposable solvable Lie algebras for which the nilradical is five-dimensional, is presented. For each class of algebra a matrix representation and a system of vector fields representation is given. As a consequence it follows that the dimension of the minimal matrix representation of any six-dimensional Lie algebra is at most six.

Keywords

Lie algebra; Lie group; Mubarakzyanov algebra; Minimal matrix representation; Vector field representation MSC Classification: 17B99

Introduction

As long ago as 1963 Mubarazyanov classified real six-dimensional indecomposable solvable Lie algebras for which the nilradical is five-dimensional [1]. This paper is frequently referred to but suffers from a number of defects. Recently Shabanskaya and Thompson [2] reconsidered Mubarazyanov's paper and made a number of improvements. Many errors, both substantive and typographical were corrected, some algebras were shown to be redundant, others missing; finally even in those algebras that were essentially correct, in many classes of algebras that depend on parameters, the parameters could be simplified and some removed entirely.

In the present article, given that Mubarazyanov's list has now been corrected, we turn to the issue of finding representations. We know of course that finding representations for solvable algebras is a completely different enterprise from the semi-simple case. In most cases the algebra will have a trivial center and so the adjoint representation furnishes a faithful representation. However, using the adjoint representation is not always satisfactory. In the first place many algebras have non-trivial centers; secondly, it frequently happens that in a class of algebras that depend on parameters the center will be non-trivial except for certain special values of the parameters. Therefore, one has to use a different representation in the exceptional case, which is an unsatisfactory state of affairs. In the representations given below we have endeavored to the extent possible to avoid such split cases.

We give two kinds of representations for the solvable algebras. First of all we consider matrix representations, in most cases as subalgebras of gl (6,R) or occasionally gl (5,R). However, there is something of a logistical problem in that it is wasteful of space to present a list of six matrices for each class of algebras. Alternatively, one could write a single matrix, M say, depending on some variables such that the span of the matrices engenders the given algebra. In this paper we go one step further and simply convert the given 5×5 or 6×6 into a single 25 or 36 row-vector, that we call the DNA vector, by using the rows of M.

In the second kind of representation we give a system of six vector fields on R6 using local coordinates (p, q, x, y, z, w), which furnish a basis for the algebra under Lie bracket. Actually, in our opinion the best way to represent a solvable algebra is to give the matrix Lie group that is engendered by exponentiating the space of matrices spanned by the matrix M referred to in the preceding paragraph. It is then a simple matter to construct the matrix Lie algebra by differentiating and evaluating at the identity. One can also easily construct a system of left or right-invariant vector fields starting from the group matrix. All of these quantities can be obtained in principle from the DNA-vector, which explains the name. We hope eventually to publish such a list of matrices in a suitable venue but again space is an issue.

There is another interesting consequence of our list of matrix representations. Following Burde [3,4] we define the invariant μ (g) to be the minimum value of p such that a Lie algebra p∈N has a representation as a subalgebra of gl (p,R). Then we can assert that μ ≤ 6 for all of the Mubarakzyanov algebras. Ado's Theorem states that there exists a finite-dimensional faithful representation as a subalgebra of gl (p,R) for some p∈N . The theorem does not give much information about the value of p but leads one to believe that p may be very large in relation to the size of n where n is the dimension of g. A little care must be exercised because there may well be inequivalent representations for which this minimum value is attained. Of course if g has a trivial center then the adjoint representation furnishes a faithful representation of g and in the notation used above. μ ≤ n. Nonetheless many algebras have non-trivial centers, nilpotent algebras for example, and then the adjoint representation is not faithful. Even if the center is trivial it could well be the case that μ(g)<n.

It has been shown in Ghanam et al. [5] that for all five-dimensional algebras μ(g) ≤ 5. If we refer Ghanam et al. and Rawashdeh and Thompson [6,7] then we can assert quite generally that for all sixdimensional algebras μ(g) ≤ 6.

Of course it is interesting to ascertain the value of μ from a theoretical point of view. However, an important practical reason is that calculations involving symbolic programs such as Maple and Mathematica use up lots of memory when storing matrices; accordingly, calculations are likely to be faster if one can represent matrix Lie algebras using matrices of a small size. In this paper the many calculations were performed with the help of Maple.

This article is organized as follows. In Table 2, we discuss the classification of the low-dimensional Lie Algebras. In Table 3, we list all the possible five-dimensional nilpotent algebras that occur as nilradicals in the Mubarakzyanov algebras. In Table 4, we reproduce mutatis mutandis the updated list of Mubarakzyanov algebras that appeared in Shabanskaya and Thompson [2]. In Table 5, we provide matrix representations for all the Mubarakzyanov algebras using the device of DNA-vectors introduced above. Finally in Table 6, we supply the list of vector field representations. In these last two Sections in the interests of efficiency we are content simply to provide the lists without any accompanying theory.

The second author acknowledges the support of the University of Toledo in the form of a summer research undergraduate fellowship.

Classification of Low-Dimensional Lie Algebras

Many authors have considered the classification problem for the low-dimensional Lie algebras. The classification of Lie algebras in two and three dimensions has been known for almost a century and dimensions four, five and six were carried out by two Russian mathematicians Morozov and Mubarakzyanov amongst others [1,8,9]. For a much more recent account the reader may also consult Basarab et al. and Campoamor [10,11].

An important inequality for a solvable Lie algebra g asserts that

image

where nil(g) denotes the nilradical of g [8]. In dimension six there are four possible values for dim(nil(g)), that is; three, four, five or six. Case three does not occur because such a Lie algebra is decomposable. Case six comprises six-dimensional nilpotent Lie algebras that were classified by Morozov [8] and in the latter reference representations were supplied [12]. As regards the classification of the five-dimensional nilradical algebras they were studied by Mubarakzyanov [1], who found ninety nine classes of such Lie algebra; his list has been updated in Shabanskaya and Thompson [2] and this class of algebras is the concern of the present article. In the last case, that is the dimension of the nilradical is four, the algebras were classified by Turkowski [13]. A preliminary list of representations for the Turkowski algebras has been carried out by Rawashdeh and Thompson [7]. More details concerning the classification of the low-dimensional Lie algebras may be found in Turkowski [14].

The Nilradical

The nilradical nil(g) of an indecomposable six-dimensional solvable Lie algebra g for which nil(g) is five-dimensional can be one of nine types. In his paper Mubarazyanov devotes a separate paragraph corresponding to each of these nine possible nilradicals.

• algebras 1 to 12 have abelian nilradical R5 (Table 1)

Algebra [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Conditions on parameters
†g6,1 e1 ae2 be3 ce4 de5 0 <|d| ≤ |c| ≤ |b| ≤ |a| ≤ 1
†g6,2 ae1 e1 + ae2 e3 be4 ce5 0 <|c| ≤ |b| ≤ 1
†g6,3 ae1 e1+ ae2 e2 + ae3 e4 be5 0 <|b| ≤ 1
†g6,4 ae1 e1+ ae2 e2 + ae3 e3 + ae4 e5  
g6,5 e1 e1 + e2 e2 + e3 e3 + e4 e4 + e5  
†g6,6 e1 ae2 e2 + ae3 be4 e4 + be5 a ≤ b
g6,7 ae1 e1+ ae2 e2 + ae3 be4 e4 + be5 a = 0, b = 1 or a = 1
†g6,8 ae1 be2 ce3 de4 + e5 −e4 + de5 0 <|c| ≤ |b| ≤ |a|
†g6,9 ae1 be2 e2 + be3 ce4−e5 e4 + ce5 a ≠0
g6,10 ae1 e1 + ae2 e2 + ae3 be4 −e5 e4 + be5  
†g6,11 e1 ae2−be3 be2 + ae3 ce4−de5 de4 + ce5 bd≠ 0
†g6,12 ae1 be2−e3 e2 + be3 e2 + be4−e5 e3 + e4 + be5 a≠0

Table 1: Lie algebras having nilradical isomorphic to R5.

• algebras 13 to 38 have nilradical isomorphic to R2 ⊕ H (Table 2)

Algebra [e1, e6]  [e2, e6]  [e3, e6]  [e4, e6]  [e5, e6] Remarks
g6,13  (a+b)e1 ae2 be3 e4 ce5 a2+b2 ≠ 0, |a| ≤ |b|, 0 <|c| ≤ 1
g6,14  (a+b)e1 ae2 be3 e4 e1+(a+b)e5 |a| ≤ |b|
g6,15 (a+1)e1 e2 + e4 ae3 + e5 e4 ae5 |a| ≤ 1
g6,16 e1 e2 + e4 e5 e1 + e4 0  
†g6,17.1 ae1 ae2 e4 δe1 e5 (δ = 0, 1) together g6,17 in [1])
g6,17.2*** e1 e2 e4 0 e1 + e5  
†g6,18 (a+1)e1 ae2 e3 + e4 e4 be5 b ≠0
g6,19 (a+1)e1 ae2 e3 + e4 e4 e1+(a+1)e5  
†g6,20 e1 0 e3 + e4 e1 + e4 be5 b ≠ 0
†g6,21 2ae1 ae2 + e3 ae3 e4 be5 0 <|b| ≤ 1
g6,22 2ae1 ae2 + e3 ae3 e4 e1 + 2ae5  
g6,23* 2δe1 δe2 + e3 δe3 + e4 δe4 εe1 + (2δ + a)e5 (δ, ε, a) = (1, 0, ≠−2) or (0, 0, 1) (6.23.1) (suspensions) or (1, 1, 0) (6.23.2)
†g6,24* 0 e3 e4 e1 e5  
g6,25 (a+1)e1 e2 ae3 be4 + e5 be5 |a| ≤ 1
g6,26 (a+1)e1 e2 ae3 (a+1)e4 +e5 e1+(a+1)e5 |a| ≤ 1
g6,27* (a+b)e1 ae2 be3 + e4 be4 + e5 δe1 + be5 a = 1, δ = 0 or a = 0, b = 1, δ = 0, 1
g6,28 2e1 e2 + e3 e3 be4 + e5 be5  
g6,29 2e1 e2 + e3 e3 2e4 + e5 e1 + 2e5  
g6,30 0 e3 0 e4 + e5 e5  
g6,31 2e1 e2 + e3 e3 + e4 e4 + e5 e5  
g6,32**** 2ae1 ae2 + e3 ae3−e2 δe1 + (2a + b)e4 ce5 (δ, b) = (0, ≠−2a) or = (1, 0), c ≠ 0, 2a + b ≥ c
†g6,32 2ae1 ae2 + e3 ae3− e2 be4 ce5 a ≥ 0, 0 <|b| ≤ |c|
g6,33**** 2ae1 ae2 + e3 ae3−e2 be4 e1 + 2ae5 b ≠ 0, b ≤ 2a
†g6,33 2ae1 ae2 + e3 ae3− e2 e1 + 2ae4 be5 a ≥ 0, b ≠ 0
g6,34* 2ae1 ae2 + e3 ae3−e2 (2a+b)e4+e5 δe1 + (2a + b)e5 a ≥ 0, δ = 0 or δ = 1, b = 0
g6,35 (a+b)e1 ae2 be3 ce4 + e5 ce5−e4 a2 + b2 ≠ 0, |a| ≤ |b|, c ≥ 0
g6,36 2ae1 ae2 + e3 ae3 be4 + e5 be5−e4 b ≥ 0
g6,37 2ae1 ae2 + e3 ae3−e2 be4 + ce5 be5−ce4 a ≥ 0, c >0
g6,38* 2ae1 ae2 + e3 ae3−e2+e4 ae4 + e5 ae5 −e4 a ≥ 0

Table 2: Lie algebras having nilradical isomorphic to H ⊕R2 : [e2, e3] = e1.

• algebras 39 to 53 have nilradical isomorphic to A4,1 R⊕ (Table 3)

Algebra [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Remarks
† g6,39 (b + 1) e1  (b + 2)e2 ae3 be4 e5 a ≠0
g6,40 (a + 1)e1 (a + 2)e2 e2 + (a + 2) e3 ae4 e5  
g6,41 (a + 1)e1 (a + 2)e2 ae3 e3 + ae4 e5  
g6,42 (a + 1)e1 (a + 2)e2 e3 ae4 e3 + e5  
g6,43 0 e2 e2 + e3 − e4 e3 + e5  
†g6,44 2e1a 3e2 ae3 e4 e4 + e5 a ≠0
g6,45 2e1 3e2 e2 + 3e3 e4 e4 + e5  
g6,46 2e1 3e2 e3 e3 + e4 e4 + e5  
†g6,47 e1 e2 ae3 εe2 + e4 0 a ≠0,ε = 0,±1
g6,48 e1 e2 e2 + e3 e4 0  
g6,49 e1 e2 0 εe2 + e4 e3 ε = 0, ±1
g6,50 e1 e2 εe2 + e3 e3 + e4 0 ε = 0, ±1
†g6,51 0 0 e3 εe2 0 ε = ±1
†g6,52 0 0 e3 εe2 e4 ε = 0, ±1

Table 3: Lie algebras having nilradical isomorphic to R⊕ A4,1: [e1, e5] = e2, [e4, e5] = e1.

• algebras 54 to 70 have nilradical isomorphic to A5,1 (Table 4)

Algebra [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Remarks
g6,53 0 0 e3 e4 -e5  
g6,54 e1 ae2 (1 −b)e3 (a −b)e4 be5 |a| ≤ 1
g6,55 e1 (1 + a)e2 (1−a)e3 e1 + e4 ae5  
g6,56* e1 0 e2 −e4 e5  
g6,57 e1 2ae2 (1 −a)e3 ae4 e4+ae5  
g6,58 3e1 2e2 e2 + 2e3 e4 e4 + e5  
g6,59* e1 0 e3 e2 δe4 δ = 0, 1
g6,60* e1 2e2 0 e1 + e4 e4 + e5  
g6,61* 2e1 0 e3 −e4 e3 + e5  
g6,62* 2e1 e2 e2 + e3 0 e3 + e5  
g6,63 e1 ae2 e3 e2 + ae4 0  
g6,64* e1 e2 εe2 + e3 e1 + e4 0 ε = ±1
g6,65** ae1+e2 ae2 (a −b)e3 + e4 (a −b)e4be5a = 1 or b = 1    
g6,66 2e1+e2 2e2 e3 + e4 e4 e3 + e5  
g6,67****   2e1+e2 2e2 e3 + e4 e4 ae4+e5 equivalent to g6,65(a = 2, b = 1)
g6,68* e1 + e2 e2 e3 + e4 e1 + e4 0  
g6,69****   e1 + e2 e2 e3 + e4 e2 + e4 0 equivalent to g6,65(b = 0)
g6,70***   ae1+e2 −e1+ae2 δe2+(a−b)e3+ e4 −e3+(a−b)e4 be5 δ = 0 (g6,70 in [1]) or δ = 1 and b = 0 (new algebra)

Table 4: Lie algebras having nilradical isomorphic to A5,1: [e3, e5] = e1 [e4, e5] = e2. In g6,54 - g6,65, a and b are used in place of λ and γ [1]

• algebras 71 to 75 have nilradical isomorphic to A5,2 (Table 5)

Algebra [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Remarks
g6,71 (a+3)e1 (a+2)e2 (a+1)e3 ae4 e5  
g6,72 4e1 3e2 2e3 e4 e4 + e5  
g6,73*** e1 e2 εe1 + e3 ae1 + εe2 + e4 0 ε = ±1, (if a = 0, then g6,73 [1])
g6,74 e1 e2 e3 e4 0  
g6,75 e1 e2 e3 e1+e4 0  

Table 5: Lie algebras having nilradical isomorphic to A5,2: [e2, e5] = e1, [e3, e5] = e2, [e4, e5] = e3.

• algebras 76 to 81 have nilradical isomorphic to A5,3 (Table 6)

Algebra [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Remarks
g6,76 (2a + 1)e1 (a + 1)e2 (a + 2)e3 e4 ae5 |a| ≤ 1
g6,77 e1 e2 2e3 εe1 + e4 0 ε = ±1
g6,78 −e1 0 e3 e3 + e4 −e5  
g6,79 3e1 2e2 e1 + 3e3 e4 + e5 e5  
g6,80*** 3ae1 + e3 2ae2 3ae3−e1 ae4−e5 e4 + ae5  
g6,81*** e3 0 −e1 εe1−e5 e4 ε = ±1
g6,80**** 2e1 e2 e3 0 e5 equivalent to g6,76(a = 0)
g6,81**** 2e1 e2 e3 0 εe3 + e5 ε = ±1 equivalent to g6,77

Table 6: Lie algebras having nilradical isomorphic to A5,3: [e2, e4] = e3, [e2, e5] = e1, [e4, e5] = e2.

• algebras 82 to 93 have nilradical isomorphic to A5,4 (Table 7)

Algebra [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Remarks
g6,82 λ = a, λ1 = b 2δe1 (δ + a)e2 (δ + b)e3 (δ −a)e4 (δ −b)e5 δ = 1, 0 ≤ a ≤ b or δ = 0, 0 ≤ a ≤ 1, b = 1
g6,83λ = a 2δe1 (δ+a)e2+ e3 (δ + a)e3 (δ −a)e4 (δ−a)e5− e4 δ = 1, 0 ≤ a or δ = 0, a = 1
g6,84 0 e2 0 −e4 e3  
g6,85λ = a 2e1 (a + 1)e2 e3 (1 −a)e4 e3 + e5 a ≥ 0
g6,86****   2e1 e2 + e3 e3 e4 e5 equivalent to g6,83 for α = 2, λ = 0
g6,87** 2e1 e2 + e5 e3 + e4 e4 e3 + e5  
g6,88* μ0 = a, ν0 = 1 αe1 (α/2 +a)e2+ e3 (α/2 +a) e3- e2 (α/2 - a) e4+ e5 (α/2-a)e5-e4 a ≥ 0
g6,89* s = a, ν0 = b 2δe1 (δ + a)e2 δe3 + be5 (δ −a)e4 δe5−be3 δ = 1, a ≥ 0 and b >0 or δ = 0 and a = 1, b >0 or a = 0, b = 1
g6,90** ν0 = a 2δe1 δe2 δe3 + ae5 e2 + δe4 δe5 −ae3 δ = 1, a ε= 0 or δ = 0, a = ±1
g6,91**,****   0 0 e5 e2 −e3 included in g6,90 as δ = 0, a = 1
g′6,92*,*** s = a, ν0 = b 2δe1 δe2 + ae4 δe3 + be5 δe4−ae2 δe5−be3 δ = 1, 0 < a ≤ |b| or δ = 0, a = 1, 0 <|b| ≤ 1 are new algebras
g6,92**** ν0 = a, μ0 = b 2δe1 δe2 + ae3 δe3−be2 δe4 + be5 δe5−ae4 equivalent to cases of g6,82, g6,83 or g6,88
g6,93** ν0 = a 2δe1 δe2 + e4 + ae5 δe3 + ae4 δe4−ae3 δe5−ae2− e3 δ = 1, a ≥ 0 or δ = 0, a = 1

Table 7 : Lie algebras having nilradical isomorphic to A5,4: [e2, e4] = e1, [e3, e5] = e1.

• algebras 94 to 98 have nilradical isomorphic to A5,5 (Table 8)

Algebra [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Remarks
g6,94 (a + 2)e1 (a + 1)e2 ae3 2e4 e5  
g6,94 (a + 2)e1 (a + 1)e2 ae3 2e4 e5  
g6,94 (a + 2)e1 (a + 1)e2 ae3 2e4 e5  
g6,97 4e1 3e2 2e3 + e4 2e4 e5  
g6,98 *e1 δe1 + e2 e3 0 δe4 δ = 0, 1

Table 8: Lie algebras having nilradical isomorphic to A5,5: [e3, e4] = e1, [e2, e5] = e1, [e3, e5] = e2.

• algebra 99 has nilradical isomorphic to A5,6 (Table 9).

Algebra [e3, e4] [e2, e5] [e3, e5] [e4, e5] [e1, e6] [e2, e6] [e3, e6] [e4, e6] [e5, e6] Remarks
g6,99 e1 e1 e2 e3 5e1 4e2 3e3 2e4 e5  

Table 9: Lie algebras having nilradical isomorphic to A5,6.

Here Rn denotes the n-dimensional abelian Lie algebra, H denotes the three-dimensional Heisenberg algebra, A4,1 and the algebras A5,n are taken from [12] and denote nilpotent algebras, 4 and 5, respectively, being the dimensions of the algebra.

Classification of the Mubarakzyanov Algebras

In this Section we give an amended version of Mubarakzyanov's list of algebras. We have simplified the notation using for the parameters a,b,c,d. Also ε can only assume the values of 0 or ± 1 and δ can only be 0 or 1 and α can only be 0 or 2 except in g6,8,8. We do not consider δ and ε to be parameters since they only assume discrete values. At the start of each table we supply the brackets for the nilradical and in the table the brackets of the basis vectors with e6. We do not highlight algebras where we merely improve the range of values of the parameters.

• Algebras below that have no asterisks are considered to be essentially correct and differ from Mubarakzyanov's algebras mutatis mudandis, that is, only by small notational differences

• Algebras that have one asterisk denote algebras in which the parameters in Mubarakzyanov's algebra can be simplified or an entry can be removed completely

• Algebras that have two asterisks denote algebras in which there is a serious computational or merely typographical mistake

• Algebras that have three asterisks denote algebras which do not appear at all in Mubarakzyanov's list

• Algebras that have four asterisks denote algebras which are redundant

• Algebras that are marked with † are suspensions in the sense of Table 3 in Shabanskaya and Thompson [2]

• Algebras that are marked with ‘ supersede a Mubarakzyanov algebra that is problematic (Tables 10 and 11).

Algebra DNA vector
g6,1 −s6, 0, 0, 0, 0, s1, 0,−as6, 0, 0, 0, s2, 0, 0,−bs6, 0, 0, s3, 0, 0, 0,−cs6, 0, s4, 0, 0, 0, 0,−ds6, s5, 0, 0, 0, 0, 0, 0
g6,2 as6,−s6, 0, 0, 0, s1, 0,−as6, 0, 0, 0, s2, 0, 0,−s6, 0, 0, s3, 0, 0, 0,−cs6, 0, s4, 0, 0, 0, 0,−ds6, s5, 0, 0, 0, 0, 0, 0
g6,3 −as6,−s6, 0, 0, 0, s1, 0,−as6,−s6, 0, 0, s2, 0, 0,−as6, 0, 0, s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0,−ds6, s5, 0, 0, 0, 0, 0, 0
g6,4 −as6,−s6, 0, 0, 0, s1, 0,−as6,−s6, 0, 0, s2, 0, 0,−as6,−s6, 0, s3, 0, 0, 0,−as6, 0, s4, 0, 0, 0, 0,−bs6, s5, 0, 0, 0, 0, 0, 0
g6,5 −s6,−s6, 0, 0, 0, s1, 0,−s6,−s6, 0, 0, s2, 0, 0,−s6,−s6, 0, s3, 0, 0, 0,−s6,−s6, s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,6 −as6,−s6, 0, 0, 0, s4, 0,−as6, 0, 0, 0, s5, 0, 0,−bs6,−s6, 0, s2, 0, 0, 0,−bs6, 0, s3, 0, 0, 0, 0,−s6, s1, 0, 0, 0, 0, 0, 0
g6,7 −bs6,−s6, 0, 0, 0, s5, 0,−bs6, 0, 0, 0,−s4, 0, 0,−as6,−s6, 0, s1, 0, 0, 0,−as6,−s6, s2, 0, 0, 0, 0,−as6, s3, 0, 0, 0, 0, 0, 0
g6,8 −ds6,−s6, 0, 0, 0, s5, s6,−ds6, 0, 0, 0, s4, 0, 0,−as6, 0, 0, s2, 0, 0, 0,−bs6, 0, s3, 0, 0, 0, 0,−cs6, s1, 0, 0, 0, 0, 0, 0
g6,9 −cs6,−s6, 0, 0, 0, s4, s6,−cs6, 0, 0, 0, s5, 0, 0,−bs6,−s6, 0, s2, 0, 0, 0,−bs6, 0, s3, 0, 0, 0, 0,−as6, s1, 0, 0, 0, 0, 0, 0
g6,10 −as6,−s6, 0, 0, 0, s1, 0,−as6,−s6, 0, 0, s2, 0, 0,−as6, 0, 0, s3, 0, 0, 0,−bs6, s6,−s4, 0, 0, 0,−s6,−bs6, s5, 0, 0, 0, 0, 0, 0
g6,11 −as6, 0, 0, 0, 0, s1, 0,−bs6,−s6, 0, 0, s2, 0, s6,−bs6, 0, 0, s3, 0, 0, 0,−cs6,−ds6, s5, 0, 0, 0, ds6,−cs6,−s4, 0, 0, 0, 0, 0, 0
g6,12 −as6, 0, 0, 0, 0, s1, 0,−bs6,−s6,−s6, 0, s3, 0, s6,−bs6, 0,−s6,−s2, 0, 0, 0,−bs6,−s6, s5, 0, 0, 0, s6,−bs6,−s4, 0, 0, 0, 0, 0, 0
g6,13 −(a + b)s6,−s3, s2, 0, 0, 2s1, 0,−as6, 0, 0, 0, s2, 0, 0,−bs6, 0, 0, s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0,−cs6, s5, 0, 0, 0, 0, 0, 0
g6,14 −(a + b)s6,−s2, s3,−2s6, 0,−2s1, 0,−as6, 0, 0, 0, s3, 0, 0,−bs6, 0, 0, s2, 0, 0, 0,−(a + b)s6, 0, s5, 0, 0, 0, 0,−s6,
s4, 0, 0, 0, 0, 0, 0
g6,15 −(a + 1)s6,−s2, s3, 0, 0,−s1, 0,−s6, 0, 0, 0, s3, 0, 0,−as6, 0, 0, as2, 0,−s6, 0,−s6, 0, s3 + s5,
0, 0,−s6, 0,−as6, s2 + as4, 0, 0, 0, 0, 0, 0
g6,16 0, 0, 0, 0,−s6,−1 2 s5, 0,−s6,−s6,−1 2 s3,−s2, s1, 0, 0,−s6,−s6, 0, s4, 0, 0, 0,−s6, 0, s2, 0, 0, 0, 0, 0,−1 2 s3, 0, 0, 0, 0, 0, 0
g6,17.1 −s6, 0, 0, 0, 0, s5, 0,−as6, 0,−s6,−s3, s1, 0, 0,−as6, 0, 0,−s3, 0, 0, 0, 0,−s6, s4, 0, 0, 0, 0, 0, s2, 0, 0, 0, 0, 0, 0
g6,17.2 0, s3, 0, 0, 0,−s4, 0, 0, 0, 0, 0,−s6, 0, 0,−s6, s3,−s6,−s1, 0, 0, 0,−s6, 0, s2, 0, 0, 0, 0,−s6,−s5, 0, 0, 0, 0, 0, 0
g6,18 −bs6, 0, 0, 0, 0, s5, 0,−s6, 0,−s6, 0, s4, 0, 0,−(a + 1)s6, 0, s3,−s1, 0, 0, 0,−s6, 0, s3, 0, 0, 0, 0,−as6, s2, 0, 0, 0, 0, 0, 0
g6,19 −(a + 1)s6, 0,−s3, s2,−s6, 2s1, 0,−s6, 0,−s6, 0, s4, 0, 0,−as6, 0, 0, s2, 0, 0, 0,−s6, 0, s3, 0, 0, 0, 0,−(a + 1)s6,
2s5, 0, 0, 0, 0, 0, 0
g6,20 −bs6, 0, 0, 0, 0, s5, 0,−s6,−s6, s2,−s3, s1, 0, 0,−s6,−s6, 0, s4, 0, 0, 0,−s6, 0, s3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,21 −2as6, as2, s2−s3, 0, 0, 2s1, 0,−as6,−s6, 0, 0, s3, 0, 0,−as6, 0, 0, as2, 0, 0, 0,−s6, 0, s5, 0, 0, 0, 0,−bs6, s4, 0, 0, 0, 0, 0, 0
g6,22 −s6, 0, 0, 0, 0, s4, 0,−2as6, s2,−s3, 2εs6, 2s1, 0, 0,−as6,−as6, 0, s3, 0, 0, 0,−as6, 0, s2, 0, 0, 0, 0,−2as6,−s5, 0, 0, 0, 0, 0, 0
g6,23.1 −2s6, 0, s2,−s3, 2s6, 2s1, 0,−s6,−s6, 0, 0, s3 + s4, 0, 0,−s6,−s6, 0, s2 + s3, 0, 0, 0,−s6, 0, s2, 0, 0, 0, 0,
−2s6,−s5, 0, 0, 0, 0, 0, 0
g6,23.2 −(2δ + a)s6, 0, 0, 0, 0, s5, 0,−2δs6, 0, s2, s2−s3, s1, 0, 0,−δs6,−s6, 0, s4, 0, 0, 0,−δs6,−s6, s3, 0, 0, 0, 0,−δs6, s2,
0, 0, 0, 0, 0, 0
g6,24 −s6, 0, 0, 0, 0, s5, 0, 0,−2s6, s2,−s3,−2s1, 0, 0, 0,−s6, 0,−s4, 0, 0, 0, 0,−s6,−s3, 0, 0, 0, 0, 0,−s2, 0, 0, 0, 0, 0, 0
g6,25 −bs6, s4, 0, 0, 0,−s5, 0, 0, 0, 0, 0,−s6, 0, 0,−(a + 1)s6, 0, s3,−s1, 0, 0, 0,−as6, 0, s3, 0, 0, 0, 0,−s6, s2, 0, 0, 0, 0, 0, 0
g6,26 −as6,−s6, 0, 0, s2, s1, 0,−as6,−s6, 0, 0, s5, 0, 0,−as6, 0, 0, s4, 0, 0, 0,−s6, 0, s2, 0, 0, 0, 0, (1 −a)s6, s3, 0, 0, 0, 0, 0, 0
g6,27 −(a + b)s6, δs6, 0,−s2,−s1, 0,−bs6,−s6, 0, s5, 0, 0,−bs6,−s6, s4, 0, 0, 0,−bs6, s3, 0, 0, 0, 0, 0
g6,28 −2s6, s2,−s3, 0, 0, 2s1, 0,−s6,−s6, 0, 0, s3, 0, 0,−s6, 0, 0, s2, 0, 0, 0,−bs6,−s6, s5, 0, 0, 0, 0,−bs6, s4, 0, 0, 0, 0, 0, 0
g6,29 −2s6, s2,−1 2 s3−s6,−s6, 0, s1, 0,−s6,−s6, 0, 0, 1 2 s3, 0, 0,−s6, 0, 0, s2, 0, 0, 0,−2s6,−s6, s5, 0, 0, 0, 0,−2s6, s4, 0,  0, 0, 0, 0, 0
g6,30 −s6, 0, 0, s4,−s5, 0, 0, s2, 0,−s1, 0, 0, 0, s2,−s3, 0, 0, 0, 0,−s6, 0, 0, 0, 0, 0
g6,31 −2s6, 0, 0, 3
2 s2,−3 2 s3−3s6, 9 2 s1, 0,−s6,−s6, 0, 0, 3 2 s5, 0, 0,−s6,−s6, 0, 3 2 s4, 0, 0, 0,−s6,−s6, 3 2 s3, 0, 0, 0, 0,−s6, 3 2  2, 0, 0, 0, 0, 0, 0
g6,32 −cs6, 0, 0, 0, 0, s5, 0,−bs6, 0, 0, 0, s4, 0, 0,−2as6,−s2, s3,−2s1, 0, 0, 0,−as6,−s6, s3, 0, 0, 0, s6,−as6, s2, 0, 0, 0, 0, 0, 0
g6,33 −bs6, 0, 0, 0, 0, s5, 0,−2as6,−s3, s2,−s6, 2s1, 0, 0,−as6, s6, 0, s2, 0, 0,−s6,−as6, 0, s3, 0, 0, 0, 0,−2as6, 2s4, 0, 0, 0, 0, 0, 0
g6,34 −2as6,−s3, s2,−δs6, 0, 2s1, 0,−as6, s6, 0, 0, s2, 0,−s6,−as6, 0, 0, s3, 0, 0, 0,−(2a + b)s6,−s6, 2s5, 0, 0, 0,
0,−(2a + b)s6, 2s4, 0, 0, 0, 0, 0, 0
g6,35 −cs6, s6, 0, 0, 0, s4,−s6,−cs6, 0, 0, 0, s5, 0, 0,−(a + b)s6,−s3, s2, 2s1, 0, 0, 0,−as6, 0, s2, 0, 0, 0, 0,−bs6, s3, 0, 0, 0, 0, 0, 0
g6,36 −cs6, s6, 0, 0, 0, s4,−s6,−cs6, 0, 0, 0, s5, 0, 0,−2as6,−s2, s3,−2s1, 0, 0, 0,−as6,−s6, s3, 0, 0, 0, 0,−as6, s2, 0, 0, 0, 0, 0, 0
g6,37 −a2s6,−s2, s3, 0, 0,−2s1, 0,−as6,−s6, 0, 0, s3, 0, s6,−as6, 0, 0, s2, 0, 0, 0,−bs6,−cs6, s5, 0, 0, 0, cs6,−bs6, s4, 0, 0, 0, 0, 0, 0
g6,38 −2as6, 0, 0,−s3,−s2,−2s1, 0,−as6,−s6,−s6, 0,−s3 + 2s5, 0, s6,−as6, 0,−s6, 2s4, 0, 0, 0,−as6, −s6,−s2, 0, 0, 0, s6,−as6, s3, 0, 0, 0, 0, 0, 0
g6,39 (b −a + 2)s6, 0, 0, 0, s3, 0, 0, s5, 0, s2, 0, 0, s6, s5,−s1, 0, 0, 0, 2s6, s4, 0, 0, 0, 0, (b + 2)s6
g6,40 0, s5, 0, s3, s2, 0, s6, s5, 0,−s1, 0, 0, 2s6, 0, s4, 0, 0, 0, (a + 2)s6, s6, 0, 0, 0, 0, (a + 2)s6
g6,41 −(2 + a)s6, s5, 0, 0, s1, s2, 0,−(a + 1)s6, 0, s5,−s4,−s1, 0, 0,−as6, s6, 0,−s3, 0, 0, 0,−as6, 0, s4, 0, 0, 0, 0,−s6, 0, 0, 0, 0, 0, 0, 0
g6,42 −s6, 0, 0, s6, 0,−s3, 0,−(a + 2)s6, 1 4 s5,−1 4 s1, 0,−s2, 0, 0,−(a + 1)s6, s4,−2s5, 3s1, 0, 0, 0,−s6,
0, s5, 0, 0, 0, 0,−as6, s4, 0, 0, 0, 0, 0, 0
g6,43 −s6,−s5,−s6, 0, s1, s2, 0, 0, 0,−s5, s4, 0, 0, 0,−s6, 0,−s6, s3, 0, 0, 0, s6, 0,−s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0−s6,−s5,−s6, 0, s1, s2, 0, 0, 0,−s5, s4, 0, 0, 0,−s6, 0,−s6, s3, 0, 0, 0, s6, 0,−s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,44 3s6,−s5, 0, 0, s1, 3s2, 0, 2s6, 0,−s5, s4 −s5, 2s1, 0, 0, as6, 0, 0, s3, 0, 0, 0, s6, s6, s4, 0, 0, 0, 0, s6, s5, 0, 0, 0, 0, 0, 0
g6,45 3s6,−s5, s6, 0, s1, 3s2, 0, 2s6, 0,−s5, s4 −s5, 2s1, 0, 0, 3s6, 0, 0, 3s3, 0, 0, 0, s6, s6, s4, 0, 0, 0, 0, s6, s5, 0, 0, 0, 0, 0, 0
g6,46 3s6,−s5, 0, 0, s1, 3s2, 0, 2s6, 0,−s5, s4 −s5, 2s1, 0, 0, s6, s6, 0, s3, 0, 0, 0, s6, s6, s4, 0, 0, 0, 0, s6, s5, 0, 0, 0, 0, 0, 0
g6,47 −s6, s5, 0,−εs6,−s1, s2, 0,−s6, 0, s5, s4,−s1, 0, 0,−as6, 0, 0, s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,48 −s6, s5,−s6, 0,−s1, s2, 0,−s6, 0, s5, s4,−s1, 0, 0,−s6, 0, 0, s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,49 0, 0, 0,−s6, 0, s3, 0,−s6,−s5, s1,−s6, 2s2, 0, 0,−s6, s4, 0, s1, 0, 0, 0, 0, 0, s5, 0, 0, 0, 0,−s6, 2εs4, 0, 0, 0, 0, 0, 0
g 6,50 −s6, s5,−εs6, 0,−s1, s2, 0,−s6, 0, s5, s4,−s1, 0, 0,−s6,−s6, 0, s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,51 −s6, 0, 0, 0, 0, s3, 0, 0, s5, 0, s4, s2, 0, 0, 0, s5, 0,−s1, 0, 0, 0, 0, 0, s4, 0, 0, 0, 0, 0, εs6, 0, 0, 0, 0, 0, 0
g6,52 −εs6, 0, 0, 0, 0, s3, 0, 0, 2s5,−s6, s4, 2s2, 0, 0, 0, s5, 0,−s1, 0, 0, 0, 0, s5, s4, 0, 0, 0, 0, 0, s6, 0, 0, 0, 0, 0, 0
g6,53 s6, 0, s5, 0,−s2, 0, s6, 0, s5,−s1, 0, 0, 0, 0, s4, 0, 0, 0, 0, s3, 0, 0, 0, 0, s6
g6,54 −as6, 0, s5, 0,−s2, 0,−s6, 0, s5,−s1, 0, 0, (b −a)s6, 0, s4, 0, 0, 0, (b −1)s6, s3, 0, 0, 0, 0, 0
g6,55 −(a + 1)s6, 0, 0, 0, s5,−1 2 s2, 0,−s6, 1 2 s3, s5, s6,−1 2 s1, 0, 0,−as6, 0, 0, s5, 0, 0, 0, (a −1)s6, 0, s3, 0, 0, 0,  ,−s6, 1 2 s4, 0, 0, 0, 0, 0, 0
g6,56 −s6, 0, s5, 0,−s1, 0, 0,−s6, s5, s2, 0, 0, 0, 0, s3, 0, 0, 0, s6,−s4, 0, 0, 0, 0, 0
g6,57 −s6, s5, 0, 0, 0,−s1, 0, (a −1)s6, 0, 0, 0, s3, 0, 0,−2as6, s5, 0, s2, 0, 0, 0,−as6, s5,−s4, 0, 0, 0, 0, 0,−s6, 0, 0, 0, 0, 0, 0
g6,58 −3s6, 0,−s5, 0, 1 2 s3, 3 2 s1, 0,−2s6,−s6,−s5, s4 −s5, 2s2, 0, 0,−2s6, 0, 0, s3, 0, 0, 0,−s6,−s6, s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,59 −s6, s5, 0, 0, 0,−s1, 0,−s6, 0, 0, 0, s3, 0, 0, 0,−s5 −s6, 0,−s2, 0, 0, 0, 0,−s5 −s6,−s4, 0, 0, 0, 0, 0,−δs5, 0, 0, 0, 0, 0, 0
g6,60 −s6, 0,−s5,−s6, s3, s1, 0,−2s6, 0,−s5, s4 + s5, 2s2, 0, 0, 0, 0, 0, 0, 0, 0, 0,−s6,−s6, s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,61 0, 0, 0, 0, s4, s2, 0, s6, 0, 0, 0, s4, 0, 0,−2s6,−s5, s3, 2s1, 0, 0, 0,−s6,−s6, s3, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,62 2s6, 0,−s5, s3 −1 2 s5, 2s1, 0, s6, s6, 1 4 s4, 3s2, 0, 0, s6,−1 2 s6, s3, 0, 0, 0, s6, s5, 0, 0, 0, 0, 0
g6,63 −as6,−s5 −s6, 0, 0, 0, s2, 0,−as6, 0, 0, 0, s4, 0, 0,−s6, s5, s3,−s1, 0, 0, 0,−s6, 0, s3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,64 −s6, 0, s5,−s6,−s3, s1, 0,−s6,−εs6, s5, s4,−s2, 0, 0,−s6, 0, 0,−s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,65 −as6, 0, s6, 0,−s4,−s2, 0, (b −a)s6, 0, s6, 0,−s4, 0, 0,−as6, 0, s3, s1, 0, 0, 0, (b −a)s6, 0, s3, 0, 0, 0, 0,−bs6, s5, 0, 0, 0, 0, 0, 0
g6,66 −2s6,−s6,−s5, 0,−s3 + s4 + s5,−s1 + 2s2, 0,−2s6, 0,−s5, s3 −s5, 2s1, 0, 0,−s6,−s6, 0, s4, 0, 0, 0,−s6,−s6, s3,
0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,68 −s6,−s6,−s5, 0,−s3 + s4, s2, 0,−s6,−s6,−s5, s3, s1, 0, 0,−s6,−s6, 0, s4, 0, 0, 0,−s6, 0, s3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,70 −as6, s6, s5, 0,−s3, as1 −s2,−s6,−as6, δs6, s5,−s4, s1 + as2 + δs3, 0, 0, (b −a)s6, s6, 0, (b −a)s3 + s4, 0, 0,−s6,
(b −a)s6, 0,−s3 + (b −a)s4, 0, 0, 0, 0,−bs6,−bs5, 0, 0, 0, 0, 0, 0
g6,71 −(a + 3)s6, s5, 0, 0,−s1, 0,−(a + 2)s6, s5, 0, s2, 0, 0,−(a + 1)s6, s5,−s3, 0, 0, 0,−as6, s4, 0, 0, 0, 0, 0
g6,72 −4s6,−s5, 0, 0, s2, 4s1, 0,−3s6,−s5, 0, s3, 3s2, 0, 0,−2s6,−s5, s4 −s5, 2s3, 0, 0, 0,−s6,−s6, s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,73 −s6,−s5,−εs6,−as6, s2 −εs4, s1, 0,−s6,−s5,−εs6, s3, s2, 0, 0,−s6,−s5, s4, s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,74 −s6, s5, 0, 0,−s2, s1, 0,−s6, s5, 0, s3,−s2, 0, 0,−s6, s5,−s4, s3, 0, 0, 0,−s6, 0,−s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,75 −s6,−s5, 0,−s6, s2, s1, 0,−s6,−s5, 0, s3, s2, 0, 0,−s6,−s5, s4, s3, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,76 −(a + 2)s6, 0, 1 2 s4, 1 2 s2,−s3, 0,−(2a + 1)s6, s5, 0,−s1, 0, 0,−(a + 1)s6, s5, s2, 0, 0, 0,−s6,−s4, 0, 0, 0, 0, 0
g6,77 −s6, 0, s5,−εs6,−s2, s1, 0,−2s6,−s4,−s2, 0,−2s3, 0, 0,−s6, s5, s4,−s2, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,78 s6, 0,−s5, 0, s2,−s1, 0,−s6,−s4, s2 −s6, 0, s3, 0, 0, 0,−s5, s4, 0, 0, 0, 0,−s6, 0, s4, 0, 0, 0, 0, s6,−s5, 0, 0, 0, 0, 0, 0
g6,79 −3s6,−s6, s4 −s5, s2, 0, 3s1 −2s3, 0,−3s6,−s4, 0, s2, 3s3, 0, 0,−2s6, s4, s4 −s5, 2s2, 0, 0, 0,−s6,
−s6, s5, 0, 0, 0, 0,−s6, s4, 0, 0, 0, 0, 0, 0
g6,81 0, s6,−s4,−εs6, s2, s1,−s6, 0, s5, s2, 0, s3, 0, 0, 0,−s4,−s5, 0, 0, 0, 0, 0,−s6, s4, 0, 0, 0, s6, 0, s5, 0, 0, 0, 0, 0, 0
g6,82 −2δs6,−s4, s2, 0, 0, s1, 0,−(δ + a)s6, 0, 0, 0, s3, 0, 0,−(δ + b)s6, 0, 0, s5, 0, 0, 0,−(δ + a)s6, 0,−s4, 0, 0, 0, 0, (b − δ)s6, s2, 0, 0, 0, 0, 0, 0
g6,83 −2δs6, s5,−s4, 0, 0, s1, 0,−(δ + a)s6, s6, 0, 0,−s3, 0, 0,−(δ + a)s6, 0, 0, s2, 0, 0, 0, (a − δ)s6,−s6,−s4, 0, 0, 0, 0, (a − δ)s6, s5, 0, 0, 0, 0, 0, 0
g6,84 0, s5, 0, s4, s1, 0, 0, s5, 0,−s3, 0, 0, 0, 0,−s6, 0, 0, 0,−s6,−s2, 0, 0, 0, 0, 0
g6,85 −2s6,−s4,−s5,−s2,−s3,−2s1, 0,−(a + 1)s6, 0, 0, 0,−s2, 0, 0,−s6, 0, s6,−s3, 0, 0, 0, (a −1)s6, 0, s4,
0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,87 −2s6, s2, s2 −s5, s2 + s3 −s5, s2 + s3 −s4 −s5, 2s1, 0,−s6,−s6, 0, 0, s4, 0, 0,−s6,−s6, 0, s3, 0, 0, 0,−s6,−s6,
s5, 0, 0, 0, 0,−s6, s2, 0, 0, 0, 0, 0, 0
g6,88 −2αs6,−s2,−s3, 0, 0,−s1, 0,−(α + a)s6, bs6, 0, 0, s4, 0,−bs6,−(α + a)s6, 0, 0, s5, 0, 0, 0,−(α + a)s6,
bs6, 0, 0, 0, 0,−bs6,−(α + a)s6, 0, 0, 0, 0, 0, 0, 0g6,88 −2αs6,−s2,−s3, 0, 0,−s1, 0,−(α + a)s6, bs6, 0, 0, s4, 0,−bs6,−(α + a)s6, 0, 0, s5, 0, 0, 0,−(α + a)s6, bs6, 0, 0, 0, 0,−bs6,−(α + a)s6, 0, 0, 0, 0, 0, 0, 0
g6,89 −2δs6, 0,−2s2,−s5,−s3,−2s1, 0,−(δ + a)s6, 0, 0, 0,−2s2, 0, 0,−(δ + a)s6, 0, 0, s4, 0, 0, 0,−δs6,−bs6,
−s3, 0, 0, 0, bs6,−δs6, s5, 0, 0, 0, 0, 0, 0
g6,90 −2δs6, s5,−s3,−s4, s2,−2s1, 0,−δs6, as6, 0, 0,−s3, 0,−as6,−δs6, 0, 0,−s5, 0, 0, 0,−δs6,−s6,−s2, 0, 0, 0, 0,
−δs6,−s4, 0, 0, 0, 0, 0, 0
g6,92 −2δs6,−s5, s3,−s4, s2, 2s1, 0,−δs6, as6, 0, 0, s3, 0,−as6,−δs6, 0, 0, s5, 0, 0, 0,−δs6, bs6, s2, 0, 0, 0,−bs6,−δs6, s4, 0, 0, 0,
0, 0, 0
g6,93 −2δs6,−s2,−s5, s4,−s3,−2s1, 0,−δs6, as6,−s6, 0, s4, 0,−as6,−δs6, 0,−s6,−s3, 0, 0, 0,−δs6, as6, s2, 0, 0, 0,−as6, −δs6, s5, 0, 0, 0, 0, 0, 0
g6,94 −(a + 2)s6, s5, 0, s3, s1, 0,−(a + 1)s6, s5, 0,−s2, 0, 0,−as6, 0, s3, 0, 0, 0,−2s6, s4, 0, 0, 0, 0, 0
g6,95 −2s6,−s5, s4, s3 −s6,−s2,−2s1, 0,−s6,−s5, 0,−s3,−s2, 0, 0, 0, 0, 0, 0, 0, 0, 0,−2s6, 0,−2s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,96 −3s6,−s5,−s4, s3 −s5, s2 −1 2 s4, 3s1, 0,−2s6,−s5,−s6, s3, 2s2, 0, 0,−s6, 0,−s6, s3, 0, 0, 0,−2s6, 0, 2s4, 0, 0, 0, 0, −s6, s5, 0, 0, 0, 0, 0, 0
g6,97 −4s6,−s5, s3, 1 2 s3 −s4, s2, 4s1, 0,−3s6, 0,−s5, s3, 3s2, 0, 0,−2s6,−s6, 0, 2s4, 0, 0, 0,−2s6, 0, 2s3, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0
g6,98 −s6,−s5 −δs6,−s4, s3,−s2,−s1, 0,−s6,−s5, 0, s3,−s2, 0, 0,−s6, 0, 0, s3, 0, 0, 0, 0,−δs6,−s5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
g6,99 −5s6,−s5,−s4, s3, s2, 5s1, 0,−4s6,−s5, 0, s3, 4s2, 0, 0,−3s6,−s5, s4, 3s3, 0, 0, 0,−2s6, 0, 2s4, 0, 0, 0, 0,−s6, s5, 0, 0, 0, 0, 0, 0

Table 10: Matrix Representations for Mubarakzyanov Algebras.

Algebra Basis of Vector Fields
g6,1 Dp,Dq,Dx,Dy,Dz,Dw+ pDp+ aqDq+ bxDx+ cyDy+ dzDz
g6,2 Dp,Dq,Dx,Dy,Dz,Dw+ (ap+ q)Dp+ aqDq+ xDx+ byDy+ czDz
g6,3 Dp,Dq,Dx,Dy,Dz,Dw+ (ap+ q)Dp+ (aq+ x)Dq+ axDx+ yDy+ bzDz+ Dw
g6,4 Dp,Dq,Dx,Dy,Dz,Dw+ (ap+ q)Dp+ (aq+ x)Dq+ (ax + y)Dx+ ayDy+ zDz+ Dw
g6,5 Dp,Dq,Dx,Dy,Dz,Dw+ (p + q)Dp+ (q + x)Dq+ (x + y)Dx+ (y + z)Dy+ zDz
g6,6 Dz,Dx,Dy,Dp,Dq,Dw+ (bp+ q)Dp+ bqDq+ (ax + y)Dx+ ayDy+ zDz
g6,7 Dz,Dx,Dy,Dp,Dq,Dw+ (bp+ q)Dp+ bqDq+ (ax + y)Dx+ ayDy+ zDz
g6,8 Dz,Dp,Dq,Dx,−Dy,Dw+ bpDp+ cqDq+ (dx + y)Dx+ (dy−x)Dy+ azDz
g6,9 Dz,Dp,Dq,Dx,Dy,Dw+ (bp+ q)Dp+ bqDq+ (ax + y)Dx+ (ay −x)Dy+ azDz+ Dw
g6,10 Dp,Dz,Dq,Dx,Dy,Dw+ (ap+ z)Dp+ bqDq+ (bx+ y)Dx+ (by −x)Dy+ azDz
g6,11 Dz,Dx,Dy,Dp,Dq,Dw+ (cp+ q)Dp+ (cq−p)Dq+ (ax + y)Dx+ (by −x)Dy+ zDz+ Dw
g6,12 Dz,Dx,Dy,Dp,Dq,Dw+ (bp+ q)Dp+ (bq−p)Dq+ (bx+ p + y)Dx+ (by + q −x)Dy+ azDz+ Dw
g6,13 2Dz,Dx+ yDz,−Dy+ xDz,Dq,Dp,Dw+ (a + b)pDp+ cqDq+ axDx+ byDy+ (a + b)zDz
g6,14 2Dz,Dx + yDz,−Dy+ xDz,Dq,Dp,Dw+ (a + b)pDp+ qDq+ axDx+ byDy+ ((a + b)z + 2p)Dz
g6,15 2Dz,Dp + qDz,−Dq+ pDz,Dx,−Dy,Dw+ pDp+ aqDq+ (p + x)Dx+ (ay + 1)qDy+ (a + 1)zDz
g6,16 Dp,Dx−yDp, 1 2 (Dy+ xDp),Dq, 1 2Dz,Dw+ (p + q)Dp+ (x + q)Dq+ xDx+ yDz
g6,17.1 Dq,Dz, zDq+ Dx,Dy,Dp,Dw+ pDp+ (y + qa)Dq+ xaDx+ zDy
g6,17.2 Dz,Dy,wDp+ Dx+ yDz,−Dp,Dw+ Dq, qDq+ yDy+ (q + z)Dz
g6,18 Dp,Dx+ yDp,−Dy+ xDp,−Dq,Dz,Dw+ (a + 1)pDp+ (y + q)Dq+ axDx+ yDy+ bzDz
g6,19 Dp,Dx+ yDp,−Dy+ xDp,−Dq,Dz,Dw+ (z + (a + 1)p)Dp+ (y + q)Dq+ axDx+ yDy+ (a + 1)zDz
g6,20 Dx,−Dq−zDx,Dz,Dy,Dp,Dw+ bpDp+ (x + y)Dx+ (y + z)Dy+ zDz
g6,21 Dq,−Dx+ yDq,−Dy−xDq,Dp,−Dz,Dw+ pDp+ 2aqDq + a(x + y)Dx+ ayDy+ bzDz
g6,22 Dq,−Dx+ yDq,−Dy−xDq,Dp,−Dz,Dw+ pDp+ (2aq −z)Dq+ a(x + y)Dx+ ayDy+ 2azDz
g6,23.1 −2Dp,Dq+ (x + q)Dp,Dx−qDp,Dy,Dz,Dw+ 2(p −z)Dp+ qDq+ (x + q)Dx+ (y + x)Dy+ 2Dz
g6,23.2 −2Dq,Dz+ (y + z)Dq,Dy−zDq,Dx,Dp, (a + 2δ)pDp+ 2δqDq+ (y + δx)Dx+ (z + δy)Dy+ zδDz+ Dw
g6,24 Dq,Dz−yDq,Dy+ zDq,Dx,Dp,Dw+ pDp+ 2xDq+ yDx+ zDy
g6,25 g6,25 −Dx,Dy+ zDx,Dz,Dp+ wDq,−Dq,Dw+ bpDp+ bqDq+ (a + 1)xDx+ ayDy+ zDz
g6,26 Dx,Dp+ qDx,−Dq,Dz,Dy,Dw+ pDp+ qaDq+ (y + (a + 1)x)Dx+ (z + (a + 1)y)Dy+ (a + 1)zDz
g6,27 Dp,−(Dq+ zDp),Dz,Dy,Dx,Dw+ ((a + b)p + δx)Dp+ (aq+ δw2 2 )Dq+ (y + bx)Dx+ (z + by)Dy+ bzDz
g6,28 −2Dx,−(Dz+ yDx),−Dy+ zDx,Dq,Dp,Dw+ (bp+ q)Dp+ bqDq+ 2xDx + (y + z)Dy+ zDz)
g6,29 −2Dp,−(Dq+ (x + 1)Dp),−Dx+ qDp,−2Dz,−2Dy,Dw+ (y + 2p)Dp+ qDq+ (1 + x + q)Dx+ (2y + z)Dy+ 2zDz
g6,30 Dx,Dz+ yDx+ wDy,−Dy,−Dp−wDq,Dq,Dw+ pDp+ qDq
g6,31 −1 2Dp,−1 2 (Dz+ yDp),−1 2 (Dy−zDp),−1 2Dx,−1 2Dq,Dw+ 2pDp+ (q + x)Dq+ (z + y)Dy+ (x + y)Dx+ zDz
g6,32 −Dz,−Dx,−Dy+ xDz,Dq,Dp,Dw+ cpDp+ bqDq+ (ax + y)Dx+ (ay −x)Dy+ ( 1 2 (x2 −y2) + 2az)Dz
g6,33 2Dz,−Dx + yDz,Dy−xDz, 2Dq,Dp,Dw+ bpDp+ 2aqDq+ (ax −y)Dx+ (x + ay)Dy+ (εq + 2az)Dz
g6,34 −2Dz,Dx+ yDz,Dy−xDz,Dq,Dp, (q + (2a + b)p)Dp+ (2a + b)qDq+ (ax −y)Dx+ (x + ay)Dy+ (−2δp + 2az)Dz–Dw
g6,35 −2Dz,Dx + yDz,Dy−xDz,Dp,Dq,Dw+ (cp−q)Dp+ (p + cq)Dq+ axDx+ byDy+ (a + b)zDz
g6,36 2Dz,−Dy+ xDz,−Dx−yDz,Dp,Dq,Dw+ (cp−q)Dp+ (p + cq)Dq+ (ax + y)Dx+ ayDy+ 2azDz
g6,37 2Dz,Dx+ yDz,−Dy+ xDz,Dp,−Dq, (ap+ q)Dp−(p −aq)Dq+ (ax + y)Dx+ (ay −x)Dy+ 2azDz+ Dw
g6,38 2Dz,Dp+ qDz,−Dq+ pDz,Dx,−Dy, (ap+ q)Dp+ (aq−p)Dq+ (p + xa+ y)Dx+ (q −x + ya)Dy+ 2zaDz+ Dw
g6,39 Dy,Dx,Dp,Dz,Dq+ yDx+ zDy,−Dw+ apDp+ qDq+ (b + 2)xDx+ (b + 1)yDy+ bzDz
g6,40 Dy,Dx,Dp,Dz,Dq+ yDx+ zDy,−Dw+ (a + 2)pDp+ qDq+ ((a + 2)x + p)Dx+ (a + 1)yDy+ azDz
g6,41 Dq,Dp,−Dx,Dy,Dz+ qDp+ yDq,Dw+ (a + 2)pDp+ (a + 1)qDq+ (ax −y)Dx+ ayDy+ zDz
g6,42 3Dz−x 4Dq,−Dq,Dp,Dy + xDz,−Dx−z 4Dq+ 2yDz,−Dw+ (p −x)Dp+ (a + 2)qDq+ xDx+ ayDy+ (a + 1)zDz
g6,43 −Dq−zDp,Dp,Dx,−Dy,Dz+ yzDp+ yDq,Dw+ (x + p)Dp+ wDq+ (z + x)Dx−yDy+ zDz
g6,44 Dp, 3Dq,Dx,Dy,Dz + yDp+ 3pDq,Dw+ (2p + 1 2 z2)Dp+ ( 12 z3 + 3q)Dq+ axDx+ (y + z)Dy+ zDz
g6,45 −(2Dq+ zDp), 3Dp, 3Dx,Dy+ zDq,Dz−qDp−yDq+ Dy,Dw+ (x + 3p)Dp+ 2qDq + 3xDx+ (y + z)Dy+ zDz
g6,46 −Dp, 3Dq,Dx,Dx+ Dy,Dz−yDp−3pDq,Dw(2p −1 2 z2)Dp+ (3q + 1 2 z3)Dq+ (x + y + z)Dx+ (y + z)Dy+ zDz
g6,47 Dq,Dp,Dx,Dy,Dz+ yDq+ qDp,Dw+ qDq+ (p + εy)Dp+ axDx+ yDy
g6,48 Dq,Dp,Dx,Dy,Dz+ qDp+ yDq,Dw+ (x + p)Dp+ qDq+ xDx+ yDy
g6,49 Dz+ xDq, 2Dq,−Dp,Dy + xDz, zDq−Dx,xDp+ (q + 2εy)Dq+ yDy+ zDz+ Dw
g 6,50 Dp,Dq,Dx,Dy,Dz+ yDp+ pDq,Dw+ pDp+ (q + εx)Dq+ (x + y)Dx+ yDy
g6,51 Dy,Dx,Dp,Dq+ wDx,Dz+ yDx+ qDy,−εDw+ pDp
g6,52 −Dx,−Dq,Dp,−Dy+ εw 2 Dq,Dz+ xDq+ yDx+ wDy,Dw+ pDp+ εy2 Dq
g6,53 Dx,Dq,Dz,Dy,Dp+ yDq+ zDx,−Dw−pDp+ yDy+ zDz
g6,54 Dx,Dp,Dz,Dy, yDp+ Dq+ zDx,Dw+ apDp+ bqDq+ xDx+ (a −b)yDy+ (1 −b)zDz
g6,55 −Dx,Dp,−2Dq−zDx,Dy,Dz+ yDp+ qDx,−Dw+ (a + 1)pDp+ (1 −a)qDq+ (x −y)Dx+ yDy+ azDz–Dw
g6,56 Dp,Dx,Dy,Dz, yDp+ Dq+ zDx, pDp+ qDq+ yDx−zDz+ Dw
g6,57 −Dp,−(2)Dx,Dq, 2Dy,−(Dz+ qDp+ yDx+ wDy),Dw+ pDp+ (1 −a)qDq+ 2axDx+ ayDy+ azDz+ Dw
g6,58 −3Dp,−2Dq,−2Dx −zDp,−Dy−zDq,−Dz+ xDp+ yDq−ωDy,Dw+ 3pDp+ (x + 2q)Dq+ 2xDx+ (y + z)Dy+ zDz
g6,59 −Dp,−Dx,Dq,−Dy,−(Dz+ qDp−yDx−hzDy),−Dw+ pDp+ qDq+ yDx+ hzDy
g6,60 Dp,Dq,Dx+ zDp,Dp+ zDq+ Dy,−Dz+ yDq+ Dy,Dw+ (y + p)Dp+ 2qDq−wDx+ (y −z)Dy+ zDz+ Dw
g6,61 −2Dx,−Dp, zDx+ Dy, zDp+ Dq,−yDx+ Dz,−qDq+ 2xDx+ (z + y)Dy+ zDz+ Dw
g6,62 Dp, 1 2Dq, 1 2 (Dx+ zDp−Dq), 1 2 (−Dy+ zDq),−Dz+ xDp−1 2Dx,−Dw+ 2pDp+ (x + q)Dq+ (x −z
2 )Dx+ zDz
g6,63 −Dz,Dp,Dy,Dq,Dx+ qDp−yDz,Dw+ (q + ap)Dp+ aqDq+ yDy+ zDz
g6,64 Dp,Dq,Dx,Dy,Dz+ xDp+ yDq,Dw+ (y + p)Dp+ (εx + q)Dq+ xDx+ yDy
g6,65 −Dy,Dp,Dx+ zDy,−Dq−zDp,Dz,−Dw+ (ap−y)Dp+ ((a −b)q −x)Dq+ (a −b)xDx+ ayDy+ bzDz
g6,66 Dp−2Dq,−2Dp,Dx + zDq+ Dy, zDp+ Dx,Dz−xDp−yDq+ Dy,
Dw+ (2p + q)Dp+ 2qDq + (x + y)Dx+ (y + z)Dy+ zDz
g6,67 −Dp−2Dq,−2Dp,−Dy −zDq−Dx,−Dx−zDp,−Dz+ xDp+ yDq−aDx,
Dw+ (2p + q)Dp+ 2qDq+ (x + y + az)Dx+ yDy+ zDz
g6,68 Dq+ Dp,Dp,Dx+ Dy,Dx+ cDq,Dz+ xDp+ yDq,Dw+ (p + q)Dp+ (cx + q)Dq+ (x + y)Dx+ yDy
g6,70 −1 1+a2Dp,−1 1+a2Dq, δ q+a2Dp+ aδ 1+a2Dq−Dx, −Dy,xDp+ yDq+ Dz,Dw+ (pa −q)Dp+ (p + aq+ δx)Dq+ ((a −b)x −y)Dx+ (x + (a −b)y)Dy+ bzDz
g6,71 Dp,Dq,Dx,Dy,−Dz+ qDp+ xDq+ yDx,Dw+ (a + 3)pDp+ (a + 2)qDq+ (a + 1)xDx+ ayDy+ zDz
g6,72 −4Dp,−(3Dq+ zDp),−(2Dx+ zDq),−(Dy+ zDx),−Dz+ qDp+ xDq+ yDx−Dy, Dw+ 4pDp+ 3qDq+ 2xDx+ (y + z)Dy+ zDz
g6,73 Dp,Dq,Dx,Dy,Dz+ qDp+ xDq+ yDx,Dw+ (p + ay + εx)Dp+ (q + εy)Dq+ xDx+ yDy
g6,74 Dp,Dq,Dx,Dy,Dz+ qDp+ xDq+ yDx,Dw+ pDp+ qDq+ xDx+ yDy
g6,75 Dp,Dq,Dx,Dy,Dz+ qDp+ xDq+ yDx+ Dz,Dw+ (y + p)Dp+ qDq+ xDx+ yDy
g6,76 −2Dp, 1 4 zDp+ 8 5 yDq−3Dx,−2Dq, 2 5xDq+ 1 2 zDx+ 1 2Dy, 1 2xDp −4yDx+ 2Dz,Dw+ (2a + 1)pDp+ (a + 2)qDq+ (a + 1)xDx+ yDy+ hzDz
g6,77 Dp,Dx+ yDq,−2Dq,Dy−xDq,Dz+ xDp+ yDx,Dw+ (ay + p)Dp+ 2qDq+ xDx+ yDy
g6,78 Dp,−Dx−zDp−yDq,−Dq,−Dy+ yzDq,Dz+ yDx+ yzDp,Dw−pDp+ (y + q)Dq+ yDy–zDz
g6,79 −3Dp,−(2Dx+ yDp+ zDq),−Dp−3Dq,−Dz−Dy+ xDq−yDx, −Dy+ xDp+ zDx,Dw+ (3p + q)Dp+ 3qDq+ 2xDx + (y + z)Dy+ zDz
g6,80 −Dp, yDp+ xDq+ Dz,Dq,−(Dx+ yDz+ 1 2 (x2 + y2)Dp), Dy−1 2 (x2 + y2)Dq,Dw+ (q + 3ap)Dp+ (3aq  −p)Dq+ (ax −y)Dx+ (x + ay)Dy+ (2az + 1 2 (x2 −y2))Dz
g6,81 −Dp, yDp+ xDq+ Dz,Dq,−( 1 2 (x2 + y2)Dp+ Dx+ yDz),Dy−1 2 (x2 + y2)Dq, Dw+ (q +  x)Dp−pDq−yDx+ xDy+ 1 2 (x2 −y2)Dz
g6,82 Dp,Dx,Dy,Dz+ xDp,Dq+ yDp,Dw+ 2apDp + (a −b)qDq+ (a + c)xDx+ (a + b)yDy+ (a −c)zDz
g6,83 Dp,Dx,−Dq,xDp+ Dy,−qDp−Dz, 2δpDp + (−x + qδ+ qa)Dq+ (δx+ xa)Dx+ (δy−ay + z)Dy+ (zδ−za)Dz+ Dw
g6,84 Dx,−Dq−pDx,Dy,Dp,Dz+ yDx+ wDy,−Dw−pDp+ qDq
g6,85 2Dp,Dq + yDp,Dx+ zDp,−Dy+ qDp,−Dz+ xDp,Dw+ 2pDp+ (a + 1)qDq+ (x −εz)Dx+ (1 −a)yDy+ zDz
g6,87 −2Dp,Dq+ Dz+ yDp,Dx+ Dy+ zDp,Dy−qDp, Dz+ Dx−xDp,Dw+ 2pDp+ qDq+ (x + z)Dx+ (y + x)Dy+ (z + q)Dz
g6,88 Dz,Dx,Dy,−Dp+ xDz,−Dq+ yDz, Dw+ ((α 2−a)p −q)Dp+ ((α 2 −a)q + p)Dq+ ((α  2 + a)x −y)Dx+ ((α 2  + a)y + x)Dy+ α 2 zDz
g6,89 1 2Dz, 1 2Dy, 1 2 (pDz−Dq),Dx+ yDz,−1 2 (Dp+ qDz),Dw+ (δp+ bq)Dp+ (δq−bp)Dq+ (δ −a)xDx+ (δ + a)yDy+ 2δzDz
g6,90 −2Dp,Dz + qDp,Dy+ (ax −y)Dp,Dq−zDp,Dx−(x + ay)Dp, Dw+ 2apDp+ qaDq+ (ax + by)Dx+ (−bx+ ay)Dy+ (−q + az)Dz
g6,91 g6,91 4Dp,−2xDp+ Dx−Dy, 2 1 2Dq, 2xDp+ Dx+ Dy, 2 3 2 qDp−2 12 Dz,Dw+ (q2 −z2)Dp+ zDq+ xDx−yDy–qDz
g6,92 −2Dz,−Dy+ xDz,−Dq+ pDz,Dx+ yDz,Dp+ qDz, Dw+ (αp −bq)Dp+ (αq + bp)Dq+ (αx −ay)Dx+ (αy + ax)Dy+ 2αzDz
g6,93 2Dz,−Dp + xDz,Dy+ qDz,−Dx−pDz,−Dq+ yDz, Dw+ (δp−aq)Dp+ (δq+ ap)Dq+ (δx+ p −ay)Dx+ (δy+ q + ax)Dy+ 2δzDz
g6,94 Dx,Dy,Dz+ pDx,−Dp,Dq+ yDx+ zDy,Dw+ 2pDp+ qDq+ (a + 2)xDx+ (a + 1)yDy+ azDz
g6,95 2Dp,−(Dq+ zDp),Dx−yDp+ zDq, 2Dy,Dz −qDp,Dw+ 2yDy + zDz+ (y + 2p)Dp+ qDq
g6,96 3Dp,−(2Dq + zDp),Dx+ yDp+ zDq,−Dq−2Dy+ xDp, Dx+ Dz−qDp+ (z −x)Dq,Dw+ 3pDp + (2q + y)Dq+ (x + z)Dx+ 2yDy+ zDz
g6,97 −4Dp,−3Dq −zDp,−(Dx+ 2Dy + xDp−zDq),−2Dx + yDp, −Dz+ qDp+ yDq,Dw+ 4pDp + 3qDq+ (y + 2x)Dx+ 2yDy+ zDz
g6,98 Dp,Dq−Dp,Dx+ zDp,Dy+ xDp,Dz+ qDp+ xDq−δwDy,Dw+ (δq+ p)Dp+ qDq+ xDx
g6,99 5Dp,−(4Dq+ zDp), 3Dx+ yDp+ zDq,−2Dy+ xDp−zDx, Dz−qDp−xDq−yDx,Dw+ 5pDp + 4qDq + 3xDx+ 2yDy+ zDz

Table 11: Vector Field Representations for Mubarakzyanov Algebras.

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