# Classification of Ordered Type Soliton Metric Lie Algebras by a Computational Approach

Authors: Hülya Kadıoğlu
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We classify some soliton nilpotent Lie algebras and possible candidates in
dimensions 8 and 9 up toisomorphy.We focus on1<2<⋯<n type of derivations, where n is the dimension of the Lie algebras. We present algorithms to generate possible  algebraic structures.


---

## Body

## 1. Introduction

In this paper, we compute and classifyn-dimensional (n=8,9) nilsoliton metric Lie algebras with eigenvalue type 1<2<⋯<n, which will be called “ordered type of Lie algebra” throughout this paper. We use MATLAB to achieve this goal. In the literature, six-dimensional nilpotent Lie algebras have been classified by algorithmic approaches [1]. In dimension seven and lower, nilsoliton metric Lie algebras have been classified [2–9]. Summary and details of some other classifications can be found in [10]. In our paper, we focus on dimensions eight and nine. We note that we have found that our algorithm gives consistent results with the literature in lower dimensions. We use a computational procedure that is similar to the one that we have used in our previous paper [4].In our previous paper, we classified all the soliton and nonsoliton metric Lie algebras where the corresponding Gram matrix is invertible and of dimensions7 and 8 up to isomorphism. If corresponding Gram matrix is invertible, then the soliton metric condition Uv=[1] has a unique solution. So in this case, it is easy to check if the algebra is soliton or not. But in noninvertible case, there is more than one solution. Therefore it is hard to guess if one of the solutions provides the soliton condition without solving Jacobi identity which is nonlinear. On the other hand, it may be easy if we can eliminate some algebras which admit a derivation D that does not have ordered eigenvalues without solving the following soliton metric condition Uv=[1]. For this, we prove that if the nilpotent Lie algebra admits a soliton metric with corresponding Gram matrix of η being noninvertible, all the solutions of Uv=[1] have a unique derivation. This theorem allows us to omit several cases that come from nonordered eigenvalues without considering Jacobi identity.This paper is organized as follows. In Section2, we provide some preliminaries that we use for our classifications. In Section 3, we give specific Jacobi identity conditions for Lie algebras up to dimension nine. This allows us to decide whether the Lie algebra has a soliton metric or not. In Section 4, we give details of our classifications with specific examples and provide algorithmic procedures. Section 5 contains our concluding remarks.


## 2. Preliminaries

Let(ημ,Q) be a metric algebra, where μ∈Λ2η⊗η*. Let B={Xi}i=1n be a Q-orthonormal basis of ημ (we always assume that bases are ordered). The nil-Ricci endomorphism Ricμ is defined as 〈RicμX,Y〉=ricμ(X,Y), where
(1)ricμ(X,Y)=-12∑i=1n‍〈[X,Xi],[Y,Xi]〉+14∑i=1n‍〈[Xi,Xj],X〉〈[Xi,Xj],Y〉
for X,Y∈η (we often write an inner product Q(·,·) as 〈·,·〉). When η is a nilpotent Lie algebra, the nil-Ricci endomorphism is the Ricci endomorphism. If all elements of the basis are eigenvectors for the nil-Ricci endomorphism Ricμ, we call the orthonormal basis a Ricci eigenvector basis.Now we define some combinatorial objects associated to a set of integer triplesΛ⊂{(i,j,k)∣1≤i,j,k≤n}. For 1≤i,j,k≤n, define 1×n row vector yijk to be ϵiT+ϵjT-ϵkt, where {ϵ}i=1n is the standard orthonormal basis for ℝn. We call the vectors in {yijk∣(i,j,k)∈Λ}  root vectors for Λ. Let y1,y2,…,ym (where m=|Λ|) be an enumeration of the root vectors in dictionary order. We define root matrixYΛ for Λ to be the m×n matrix whose rows are the root vectors y1,y2,…,ym. The Gram matrixUΛ for Λ is the m×m matrix defined by UΛ=YΛYΛT; the (i,j) entry of UΛ is the inner product of the ith and jth root vectors. It is easy to see that U is a symmetric matrix. It has the same rank as the root matrix; that is, Rank(UΛ)=Rank(YΛ). Diagonal elements of U are all three, and the off-diagonal entries of U are in the set {-2,-1,0,1,2}. For more information, see [11]. Let D have distinct real positive eigenvalues, and let Λ index the structure constants for η with respect to eigenvector basis B. If (i1,j1,k1)∈Λ and (i2,j2,k2)∈Λ, then 〈yi1,j1k1,yi2,j2k2〉≠2. Thus U does not contain two as an entry [4].Lemma 1.
Let(η,Q) be an n-dimensional inner product space, and let μ be an element of Λ2η*⊗η. Suppose that ημ admits a symmetric derivation D having n distinct eigenvalues 0<λ1<λ2<⋯<λn with corresponding orthonormal eigenvectors X1,X2,…,Xn. Let αijk denote the structure constants for η with respect to the ordered basis B={Xi}i=1n. Let 1≤i<j≤n. Then(1)
if there is somek∈{1,2,…,n} such that λk=λi+λj, then [Xi,Xj] is a scalar multiple of Xk; otherwise Xi and Xj commute;(2)
αijk≠0 if and only if Xk∈[ημ,ημ].Theorem 2 (see [11]).
Letη be a vector space, and let B=Xii=1n be a basis for η. Suppose that a set of nonzero structure constants αi,jk relative to B, indexed by Λ, defines a skew symmetric product on η. Assume that if (i,j,k)∈Λ, then i<j<k. Then the algebra is a Lie algebra if and only if whenever there exists m so that the inner product of root vectors 〈yijl,ylkm〉=-1 for triples (i,j,l) and (l,k,m) or (k,l,m) in Λ, the equation
(2)∑s<m‍αi,jsαs,km+αj,ksαs,im+αk,isαs,jm=0
holds. Furthermore, a term of form αi,jlαl,km is nonzero if and only if 〈yi,jl,yl,km〉=-1Theorem 3 (see [11]).
Let(ημ,Q) be a metric algebra and B={Xi}i=1n a Ricci eigenvector basis for ημ. Let Y be the root matrix for ημ. Then the eigenvalues of the nil-Ricci endomorphism are given by
(3)RicμB=-12YTv,
where v=[α2].Theorem 4 (see [4, 11]).
Let(η,Q) be a nonabelian metric algebra with Ricci eigenvector basis B. The following are equivalent. (1)
(ημ,Q) satisfies the nilsoliton condition with nilsoliton constant β.(2)
The eigenvalue vectorVD for D=Ric-βId with respect to B lies in the kernel of the root matrix for (ημ,Q) with respect to B.(3)
For noncommuting eigenvectorsX and Y for the nil-Ricci endomorphism with eigenvalues κX and κY, the bracket [X,Y] is an eigenvector for the nil-Ricci endomorphism with eigenvalue κX+κY-β.(4)
β=yijkRic for all (i,j,k) in Λ(ημ,B).Theorem 5 (see [4]).
Letη be an n-dimensional nonabelian nilpotent Lie algebra which admits a derivation D having distinct real positive eigenvalues. Let B be a basis consisting of eigenvectors for the derivation D, and let Λ index the nonzero structure constants with respect to B. Let U be the m×m Gram matrix. If U is invertible, then the following hold: (i)
|Λ|≤n-1;(ii)
if(i1,j1,k1)∈Λ and (i2,j2,k2)∈Λ, then 〈yi1,j1k2,yi2,j2k2〉≠-1.


## 3. Theory

This section provides some theorems and their proofs that allow us to consider fewer cases for our algoritm. The following theorem gives a pruning method while Gram matrix is noninvertible.Theorem 6.
Letη be an n-dimensional nilsoliton metric Lie algebra, and U the corresponding Gram matrix which is noninvertible. Then Ker(YT)=Ker(U). Furthermore all of the solutions of Uv=[1] correspond to a unique derivation.Proof .
Since rank of a matrix is equal to the rank of its Gram matrix, thenp=Rank(Y)=Rank(U). Let U:ℝp→ℝp and YT:ℝp→ℝn denote the linear functions (with respect to the standard basis) that correspond to the Gram matrix U and the transpose of the root matrix Y, respectively. Since Rank(U)=Rank(Y)=Rank(YT) and by rank-nullity theorem, we have
(4)Ker(U)=Ker(YT).
Let v be a particular solution and v0 the last column of reduced row echelon matrix [U,[1]]. Then v0 is also a solution of Uv0=[1]. Therefore U(v-v0)=0; that is, (v-v0)∈Ker(U). Using (4), then (v-v0)∈Ker(YT). For the solution v0, suppose that we denote D0 for the Nikolayevsky derivation, Ric0 for the Ricci tensor, and β0 for the soliton constant. Then using (3), we have
(5)(Ric-Ric0)=12Yt(v-v0)=0.
Then Ric=Ric0. Using Theorem 4, we have β=β0, which implies that D=D0.Lemma 7.
If nilsoliton metric Lie algebraη has ordered type of derivations 1<2<⋯<n, then its index set Λ consists of triples (i,j,i+j).Proof.
IfVD=(λ1,λ2,…,λn)T is the eigenvalue vector of D with eigenvector basis B={Xi}i=1n for η, then by Theorem 4, VD lies in the kernel of Y. Thus for each element (i,j,k)∈Λ, λi+λj-λk=0; that is, λi+λj=λk. By Lemma 1, [Xi,Xj]=λXk for some λ∈ℝ. Since λi=i for all i∈{1,2,…,n}, k=i+j and [Xi,Xj]=λXi+j. Hence, the index set for ordered type of derivations is of form (i,j,i+j).The next corollary describes the index triples(i,j,k) and the Jacobi identity for algebras with ordered type of derivations.Corollary 8.
The algebraη is a Lie algebra if and only if for all pairs of form (i,j,l) and (l,k,m) or (i,j,l) and (k,l,m) in ΛB with k∉{i,j} and for all m≥max{i+3,j+2,5}, the following equation holds:
(6)∑3≤s<m,s∉{i,j,k}‍αi,jsαs,km+αj,ksαs,im+αk,isαs,jm=0.
If in additionλi=i for i=1,…,n, then the algebra η is a Lie algebra if and only if for all pairs of form (i,j,i+j) and (i+j,k,i+j+k) or (i,j,i+j) and (k,i+j,i+j+k) in ΛB with k∉{i,j,i+j} and for all m=i+j+k≥max{2i+2,j+2,6}, the equation
(7)∑4≤s<m,s∉{i,j}αi,jsαs,km+αj,ksαs,im+αk,isαs,jm=0
holds.Proof.
By Theorem 7 of [11], the algebra ημ defined by μ is a Lie algebra if and only if whenever there exists m so that 〈yijl,ylkm〉=-1 for triples (i,j,l) and (l,k,m) or (k,l,m) in ΛB,  (2)
(8)∑s<m‍αi,jsαs,km+αj,ksαs,im+αk,isαs,jm=0
holds. Furthermore, if i, j, and k are distinct, the product αijlαlkm is nonzero if and only if 〈yijl,ylkm〉=-1.
Suppose that〈yijl,ylkm〉=-1 for (i,j,l)∈ΛB and (l,k,m) or (k,l,m) in ΛB. By definition of ΛB, we have i<j. By Lemma 1, j<l, l<m, and k<m. Since i<j<l<m, we know that m≥i+3. Similarly, j<l<m implies that j+2≤m. If i=k or j=k, then 〈yijl,ylkm〉=0, and so i,j, and k must be distinct. Since i,j,k, and l are all distinct and less than m, we know that m≥5. Thus an expression of form
(9)αi,jsαs,km+αj,ksαs,im+αk,isαs,jm
is nonzero only if m≥max{i+3,j+2,5}, and k∉{i,j}.
Suppose thatλi=i, and (i,j,l) and (l,k,m) are in the index set. Then from Lemma 7, l=i+j which implies that m=i+j+k. We know that 1≤i<j<l<m. Then, since j≥i+1 and k≥1, we have 2i+2≤i+j+k=m. Since 2i+2≤m, m=5 implies that i=1. So there is no possible (i,j,k), where all i,j,k are distinct and i<j<m with i+j+k. Thus if m=5, then 〈yijl,ylkm〉≠-1. Therefore, if λi=i, an expression of form
(10)αi,jsαs,km+αj,ksαs,im+αk,isαs,jm
is nonzero only if m≥max{2i+2,j+2,6}, and k∉{i,j}.
By Lemma1, X1 and X2 are in [ημ,ημ]⊥, and so αrts≠0 which implies that s≥3 and r≠s, t≠s. Therefore all expressions in (10) with s<3 or s∈{i,j,k} are identically zero and may be omitted from the summation for any m.The next corollary describes some equations in the structure constants of a nilpotent metric Lie algebra that are equivalent to the Jacobi identity. Each of the termsαijsαskm in the following equations corresponds to each of -1 entry in the Gram matrix U. Therefore, the following equations are useful for noninvertible case since there is no -1 entry in the Gram matrix for the invertible case.Corollary 9.
Let(ημ,〈⋯,⋯〉) be an n-dimensional inner product space where n≤9, and, μ be an element of Λ2η*⊗η. Suppose that the algebra ημ defined by μ admits a symmetric derivation D having n eigenvalues 1<2<⋯<n with corresponding orthonormal eigenvectors X1,X2,…,Xn. αi,jk denote the structure constants for η with respect to the ordered basis B={Xi}i=1n, and let λB index the nonzero structure constants as defined in (2). The algebra ημ is a Lie algebra if and only if
(11)α134α426+α235α516=0,(12)α123α3,47-α145α527-α246α617=0,(13)α123α358+α145α538+α156α628+α257α718+α347α718=0,(14)α123α369+α134α459-α235α459-α246α369-α347α279-α358α189=0
holds.Proof.
Following Lemma7, the index set consists of elements of form (i,j,i+j). Therefore the number s equals to i+j in the expression αi,jsαs,km. From the previous corollary,
(15)αi,jsαs,km+αj,ksαs,im+αk,isαs,jm
is nonzero only if m≥6, and k∉{i,j}. Also k≠i+j; otherwise αs,km=0.
Ifm=6, 2i+2≤m implies that i≤2; that is, possible numbers for “i” are 1 and 2. Possible and not possible (i,j,k) triples, which are being used in
(16)∑3≤s<m,s∉{i,j,k}αi,jsαs,km+αj,ksαs,im+αk,isαs,jm=0
and where i+j+k=6, are illustrated in Table 1. The notations in the table are as follows ✓:=yes; -=no; PT= possible triple.
In the case (a),i+j=k, and then it is not a possible triple. In the case (b), i<j, i,j,k are distinct, and k≠i+j. So it is a possible triple. In the case (c), i=k, and so it is not a possible triple. In the case (d), i<j, and all i,j,k are distinct as i+j≠k. Thus it is a possible triple. In the case (e), k is not a natural number, and so it is not a possible triple. Therefore only possible (i,j,k) triples are (1,3,2) and (2,3,1). Triples (1,3,2) and (2,3,1) correspond to nonzero products α134α426 and α235α5,16, respectively. Using the skew-symmetry, (2) turns into the following equation:
(17)α134α246+α235α156=0,
which gives (11). Using the same procedure for m=7, possible (i,j,k) triples are (1,2,4),(1,4,2) and (2,4,1), which correspond to nonzero α123α347, α145α527, and α246α617 respectively. Therefore (12) is obtained. Equations (13) and (14) can be obtained by the same way.Table 1
Possible(i,j,k) for m=6.
Case
i
j
k
PT
(a)
1
2
3
—
(b)
1
3
2
✓
(c)
1
4
1
—
(d)
2
3
1
✓
(e)
2
4
0
—As an illustration, we show how to use the results of this section in the following example.Example 10.
Letη be an 8-dimensional algebra with nonzero structure constants relative to eigenvector basis B indexed by
(18)Λ={(1,2,3),(1,3,4),(1,4,5),(1,6,7),(2,3,5),(2,6,8),(3,4,7),(3,5,8)}.
Computation shows that the structure vector[α2] is a solution to Uv=[1]8×1 if and only if it is of form
(19)[α2]=((α123)2(α134)2(α145)2(α167)2(α235)2(α268)2(α347)2(α358)2)=(-31710171517-817-517121771700)+X·(10-100-1010)+Y·(1-1-111-1-101).
Equation (13) from the previous corollary leads
(20)(-317+X+Y)·Y=(1517-X-Y)·Y⟹X+Y=917.
Moreover, using (12), we find that X=3/17 and Y=6/17, which means that (α167)2=-2/17. Thus η is not a Lie algebra.


## 4. Algorithm and Classifications

In this section, we describe our computational procedure and give the results in dimensions 8 and 9.


### 4.1. Algorithm

Now we describe the algorithm. The following algorithm can be used for both invertible and noninvertible cases.Input. The input is the integer n which represents the dimension.Output.The output is two 0-1 matrices Wsoliton and Uninv listing characteristic vectors for index sets Λ of Θn. The matrix Wsoliton has as its rows all possible characteristic vectors for canonical index sets Λ for nilpotent Lie algebras of dimension n with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix U is invertible. The matrix Uninv has as its rows all possible characteristic vectors for canonical index sets Λ for nilpotent Lie algebras of dimension n with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix U is noninvertible. In the dimensions 8 and 9, there is no example for invertible case. Thus Wsoliton is an empty matrix. Therefore we give the algorithm for the noninvertible case.Algorithm for the Noninvertible Case. Consider the following.(i)
Enter the dimensionn.(ii)
Compute the matrixZn.(iii)
Compute the matrixW.(iv)
Delete all rows ofW containing abelian factor which is the row that represents direct sums of Lie algebras.(v)
Remove all rows ofW such that the canonical Gram matrix U associated to the index set Λ is invertible.(vi)
Define eigenvalue vectorvD=(1,2,3,…,n)T in dimension n.(vii)
Remove all rows ofW if v(i)=v0(i)≤0 where v is the general solution of UΛv=[1]m×1 and v0 is the vector that we have defined in the proof of Theorem 6.(viii)
Remove all the rows ofW such that the corresponding algebra does not have a derivation of eigenvalue type 1<2<⋯<n.(ix)
Remove all the rows ofW such that the corresponding algebra does not satisfy Jacobi identity condition, which is obtained in Corollary 9.After this process, we solve nonlinear systems which follow from Jacobi  identity. In order to see how the algorithm works, we give the following example forn=6.Example 11.
Letn=6. Then
(21)Θ6={(1,2,3),(1,3,4),(1,4,5),(1,5,6),(2,3,5),(2,4,6)}.
So, matrix Z6 is 6×3 of form
(22)Z6=(123134145156235246).
Since |Θ6|=6, the matrix W is of size 26×6 as follows
(23)WΘ=(000000000001000010⋮⋮⋮⋮⋮⋮111111).
The first row of Wθ represents empty matrix, row two represents the subset {(2,4,6)} of Θ6, and so forth. Eliminating rows that represent direct sums, we have 33 rows in W matrix. Therefore none of the rows of W corresponds to Lie algebras that can be written as direct sums. These algebras correspond both to invertible and noninvertible Gram matrices. There is no example for the invertible case. For the noninvertible case, there is one ordered type nilsoliton metric Lie algebra η. Let B be the eigenvector basis for η, whose nonzero structure constants are indexed by
(24)Λη={(1,2,3),(1,3,4),(1,4,5),(1,5,6),(2,3,5),(2,4,6)}.
Computation shows that the structure vector [α2] is a solution to Uv=[1]6×1 if and only if it is of form
(25)[α2]=((α123)2(α134)2(α145)2(α156)2(α235)2(α246)2)=1143(212550)+t143(010-1-11).
By Corollary 9, η satisfies (11). Solving the equation for t, we find that
(26)(α134α246)2=(α235α156)2,(11+t)t=(55-t)2,(11+t)t=25.
After rescaling and solving for structure constants from [α2], we see that letting
(27)[X1,X2]=22X3,[X1,X3]=6X4,[X1,X4]=22X5,[X1,X5]=30X6,[X2,X3]=30X5,[X2,X4]=5X6
defines a nilsoliton metric Lie algebra, previously found in [3].


### 4.2. Classifications

Classification results for dimensions 8 and 9 appear in Tables2 and 3, respectively. We use vector notations to represent Lie algebra structures. For example, the list
(28)(0,0,0,0,45·23,14·24,91·25+91·34,136·17+29·26+14·35,104·18+4·27)
in the first row of Table 3 is meant to encode the metric Lie algebra (n,〈·,·〉) with orthonormal basis B={Xi}i=18 and bracket relations
(29)[X2,X3]=45X5,[X2,X4]=14X6,[X2,X5]=91X7,[X3,X4]=91X7,[X1,X7]=136X8,[X2,X6]=29X8,[X3,X5]=14X8,[X1,X8]=104X9,[X2,X7]=4X9.Table 2
8-dimensional nilsoliton metric Lie algebras.
Lie bracket
Index
Nullity
1
(
0,0
,
274
/
2223
·
12,99/764·13,527/8179·14+1532/9311·23,101/2154·15 +250/4199·24,150/3151·16+110/367·25+110/367·34,7/17·17)
6
3
2
(
0,0
,
82
/
7253
·
12
,
4
/
34
·
13
,
97
/
299
·
14
+
6
/
34
·
23
,
1
/
34
·
15
+
9
/
34
·
24
,
10
/
34
·
16
+
1998
/
12097
·
25
+
397
/
13917
·
34
,
728
/
2883
·
17
+
637
/
4000
·
35
)
6
4
3
(
0,0
,
163
/
702
·
12
,
263
/
2572
·
13
,
343
/
1334
·
14
+
547
/
2871
·
23
,
175
/
2358
·
15
+
160
/
1149
·
24
,
388
/
2509
·
16
+
518
/
6185
·
25
,
43
/
13870
·
17
+
915
/
2239
·
35
)
6
3Table 3
The 9-dimensional nilsoliton metric Lie algebras of nullities 1 and 2.
Lie bracket
Index
Nullity
1
(
0,0
,
0,0
,
45
·
23
,
14
·
24
,
91
·
25
+
91
·
34
,
136
·
17
+
29
·
26
+
14
·
35
,
104
·
18
+
4.27)
4
1
2
(0,0,1.12,0,1.14+3·23,0,6·25+6·34,8·17+1.26+2·35,6·18+2·27)
5
2
3
(0,0,21·12,0,21·14+39·23,0,13·16+70·25-70·34,88·17+42·35,
65·18+39·27)
5
2


#### 4.2.1. Candidates of Nilsoliton Metrics

Table5 illustrates how many possible candidates of Lie algebras appear in dimension 9 up to the nullity of its Gram matrix. The algebras illustrated in Table 4 are possible candidates of nilsoliton metric Lie algebras with ordered type of derivations in dimension 8. Here, as an example we give potential Lie algebra structures when the nullity of their corresponding Gram matrices are 3, 6 and 8 in Tables 6, 7, and 8 respectively for dimension nine.Table 4
The 8-dimensional nilsoliton metric Lie algebra candidates.
Lie bracket
Index
Nullity
1
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35)
6
4
2
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35)
6
4
3
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26)
6
4
4
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35)
6
5Table 5
Number of 9-dimensional nilsoliton metric Lie algebra candidates.
Nullity
Number of Lie algebras
3
98
4
81
5
45
6
22
7
7
8
1Table 6
The 9-dimensional candidates of nullities 6 and 8.
Lie bracket
Index
Nullity
1
(0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26  + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
6
6
2
(0, 0, 1.12, 1.13, 1.14, 1.15, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
3
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
4
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45)
7
6
5
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45)
7
6
6
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45)
7
6
7
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36)
7
6
8
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.36 + 1.45)
7
6
9
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
10
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45)
7
6
11
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45)
7
6
12
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36)
7
6
13
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45)
7
6
14
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45)
7
6
15
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45)
7
6
16
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36)
7
6
17
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45)
7
6
18
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45)
7
6
19
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36)
7
6
20
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45)
7
6
21
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36)
7
6
22
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27)
7
6
23
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
8Table 7
The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.
Lie bracket
Index
Nullity
1
(0, 0, 0, 0,  1.23,  1.24,  1.25  + 1.34,  1.17 +  1.26 +  1.35,  1.18,  1.27  + 1.36  + 1.45)
4
3
2
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
3
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.35,  1.18 +  1.27 +  1.45)
5
3
4
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.35,  1.18 +  1.27 +  1.36)
5
3
5
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.36 +  1.45)
5
3
6
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
7
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.36)
5
3
8
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.45)
5
3
9
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36)
5
3
10
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27)
5
3
11
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36 +  1.45)
5
3
12
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27 +  1.45)
5
3
13
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27 +  1.36)
5
3
14
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
15
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
5
3
16
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
5
3
17
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
18
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
19
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.45)
5
3
20
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36)
5
3
21
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27)
5
3
22
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
23
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
5
3
24
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
5
3
25
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
5
3
26
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
5
3
27
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
5
3
28
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
5
3
29
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
5
3
30
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
4
3
31
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
4
3
32
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
4
3
33
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.36)
4
3
34
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
4
3
35
(0, 0, 1.12,  1.13,  1.23,  0,  1.16 +  1.25,  1.17 +  1.26 +  1.35,  1.27 +  1.36 +  1.45)
4
3
36
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.45)
5
3
37
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.36)
5
3
38
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.27 +  1.36 +  1.45)
5
3
39
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.45)
5
3
40
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.36)
5
3
41
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36 +  1.45)
5
3
42
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27 +  1.45)
6
3
43
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27 +  1.36)
6
3
44
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
6
3
45
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
6
3
46
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
6
3
47
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
6
3
48
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
6
3
49
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.45)
5
3
50
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.36)
5
3
51
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.45)
5
3
52
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.36)
5
3
53
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.45)
5
3
54
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.36)
5
3
55
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27)
5
3
56
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.27 +  1.36 +  1.45)
5
3
57
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.45)
5
3
58
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.27 +  1.36 +  1.45)
5
3
59
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.45)
5
3
60
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
6
3
61
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
6
3
62
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
6
3
63
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
6
3
64
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
6
3
65
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
6
3
66
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.36 +  1.45)
6
3
67
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.45)
6
3
68
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.45)
6
3Table 8
The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.
Lie bracket
Index
Nullity
69
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.45)
6
3
70
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36)
6
3
71
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27)
6
3
72
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.45)
6
3
73
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18)
6
3
74
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.36 +  1.45)
6
3
75
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.45)
6
3
76
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.36)
6
3
77
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
6
3
78
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
6
3
79
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.27 +  1.36 +  1.45)
7
3
80
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.45)
7
3
81
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.36)
7
3
82
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.45)
7
3
83
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.36)
7
3
84
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.45)
7
3
85
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.36)
7
3
86
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27)
7
3
87
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.27 +  1.36 +  1.45)
6
3
88
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.45)
6
3
89
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.36)
6
3
90
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.45)
7
3
91
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36)
7
3
92
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27)
7
3
93
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18)
7
3
94
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.34,  1.17 +  1.26,  1.18)
7
3
95
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17,  1.18 +  1.45)
7
3
96
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17,  1.18 +  1.36)
7
3
97
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17 +  1.35,  1.18)
7
3
98
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25 +  1.34,  1.17,  1.18)
7
3


## 4.1. Algorithm

Now we describe the algorithm. The following algorithm can be used for both invertible and noninvertible cases.Input. The input is the integer n which represents the dimension.Output.The output is two 0-1 matrices Wsoliton and Uninv listing characteristic vectors for index sets Λ of Θn. The matrix Wsoliton has as its rows all possible characteristic vectors for canonical index sets Λ for nilpotent Lie algebras of dimension n with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix U is invertible. The matrix Uninv has as its rows all possible characteristic vectors for canonical index sets Λ for nilpotent Lie algebras of dimension n with ordered type nonsingular nilsoliton derivation whose canonical Gram matrix U is noninvertible. In the dimensions 8 and 9, there is no example for invertible case. Thus Wsoliton is an empty matrix. Therefore we give the algorithm for the noninvertible case.Algorithm for the Noninvertible Case. Consider the following.(i)
Enter the dimensionn.(ii)
Compute the matrixZn.(iii)
Compute the matrixW.(iv)
Delete all rows ofW containing abelian factor which is the row that represents direct sums of Lie algebras.(v)
Remove all rows ofW such that the canonical Gram matrix U associated to the index set Λ is invertible.(vi)
Define eigenvalue vectorvD=(1,2,3,…,n)T in dimension n.(vii)
Remove all rows ofW if v(i)=v0(i)≤0 where v is the general solution of UΛv=[1]m×1 and v0 is the vector that we have defined in the proof of Theorem 6.(viii)
Remove all the rows ofW such that the corresponding algebra does not have a derivation of eigenvalue type 1<2<⋯<n.(ix)
Remove all the rows ofW such that the corresponding algebra does not satisfy Jacobi identity condition, which is obtained in Corollary 9.After this process, we solve nonlinear systems which follow from Jacobi  identity. In order to see how the algorithm works, we give the following example forn=6.Example 11.
Letn=6. Then
(21)Θ6={(1,2,3),(1,3,4),(1,4,5),(1,5,6),(2,3,5),(2,4,6)}.
So, matrix Z6 is 6×3 of form
(22)Z6=(123134145156235246).
Since |Θ6|=6, the matrix W is of size 26×6 as follows
(23)WΘ=(000000000001000010⋮⋮⋮⋮⋮⋮111111).
The first row of Wθ represents empty matrix, row two represents the subset {(2,4,6)} of Θ6, and so forth. Eliminating rows that represent direct sums, we have 33 rows in W matrix. Therefore none of the rows of W corresponds to Lie algebras that can be written as direct sums. These algebras correspond both to invertible and noninvertible Gram matrices. There is no example for the invertible case. For the noninvertible case, there is one ordered type nilsoliton metric Lie algebra η. Let B be the eigenvector basis for η, whose nonzero structure constants are indexed by
(24)Λη={(1,2,3),(1,3,4),(1,4,5),(1,5,6),(2,3,5),(2,4,6)}.
Computation shows that the structure vector [α2] is a solution to Uv=[1]6×1 if and only if it is of form
(25)[α2]=((α123)2(α134)2(α145)2(α156)2(α235)2(α246)2)=1143(212550)+t143(010-1-11).
By Corollary 9, η satisfies (11). Solving the equation for t, we find that
(26)(α134α246)2=(α235α156)2,(11+t)t=(55-t)2,(11+t)t=25.
After rescaling and solving for structure constants from [α2], we see that letting
(27)[X1,X2]=22X3,[X1,X3]=6X4,[X1,X4]=22X5,[X1,X5]=30X6,[X2,X3]=30X5,[X2,X4]=5X6
defines a nilsoliton metric Lie algebra, previously found in [3].


## 4.2. Classifications

Classification results for dimensions 8 and 9 appear in Tables2 and 3, respectively. We use vector notations to represent Lie algebra structures. For example, the list
(28)(0,0,0,0,45·23,14·24,91·25+91·34,136·17+29·26+14·35,104·18+4·27)
in the first row of Table 3 is meant to encode the metric Lie algebra (n,〈·,·〉) with orthonormal basis B={Xi}i=18 and bracket relations
(29)[X2,X3]=45X5,[X2,X4]=14X6,[X2,X5]=91X7,[X3,X4]=91X7,[X1,X7]=136X8,[X2,X6]=29X8,[X3,X5]=14X8,[X1,X8]=104X9,[X2,X7]=4X9.Table 2
8-dimensional nilsoliton metric Lie algebras.
Lie bracket
Index
Nullity
1
(
0,0
,
274
/
2223
·
12,99/764·13,527/8179·14+1532/9311·23,101/2154·15 +250/4199·24,150/3151·16+110/367·25+110/367·34,7/17·17)
6
3
2
(
0,0
,
82
/
7253
·
12
,
4
/
34
·
13
,
97
/
299
·
14
+
6
/
34
·
23
,
1
/
34
·
15
+
9
/
34
·
24
,
10
/
34
·
16
+
1998
/
12097
·
25
+
397
/
13917
·
34
,
728
/
2883
·
17
+
637
/
4000
·
35
)
6
4
3
(
0,0
,
163
/
702
·
12
,
263
/
2572
·
13
,
343
/
1334
·
14
+
547
/
2871
·
23
,
175
/
2358
·
15
+
160
/
1149
·
24
,
388
/
2509
·
16
+
518
/
6185
·
25
,
43
/
13870
·
17
+
915
/
2239
·
35
)
6
3Table 3
The 9-dimensional nilsoliton metric Lie algebras of nullities 1 and 2.
Lie bracket
Index
Nullity
1
(
0,0
,
0,0
,
45
·
23
,
14
·
24
,
91
·
25
+
91
·
34
,
136
·
17
+
29
·
26
+
14
·
35
,
104
·
18
+
4.27)
4
1
2
(0,0,1.12,0,1.14+3·23,0,6·25+6·34,8·17+1.26+2·35,6·18+2·27)
5
2
3
(0,0,21·12,0,21·14+39·23,0,13·16+70·25-70·34,88·17+42·35,
65·18+39·27)
5
2


### 4.2.1. Candidates of Nilsoliton Metrics

Table5 illustrates how many possible candidates of Lie algebras appear in dimension 9 up to the nullity of its Gram matrix. The algebras illustrated in Table 4 are possible candidates of nilsoliton metric Lie algebras with ordered type of derivations in dimension 8. Here, as an example we give potential Lie algebra structures when the nullity of their corresponding Gram matrices are 3, 6 and 8 in Tables 6, 7, and 8 respectively for dimension nine.Table 4
The 8-dimensional nilsoliton metric Lie algebra candidates.
Lie bracket
Index
Nullity
1
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35)
6
4
2
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35)
6
4
3
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26)
6
4
4
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35)
6
5Table 5
Number of 9-dimensional nilsoliton metric Lie algebra candidates.
Nullity
Number of Lie algebras
3
98
4
81
5
45
6
22
7
7
8
1Table 6
The 9-dimensional candidates of nullities 6 and 8.
Lie bracket
Index
Nullity
1
(0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26  + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
6
6
2
(0, 0, 1.12, 1.13, 1.14, 1.15, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
3
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
4
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45)
7
6
5
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45)
7
6
6
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45)
7
6
7
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36)
7
6
8
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.36 + 1.45)
7
6
9
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
10
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45)
7
6
11
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45)
7
6
12
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36)
7
6
13
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45)
7
6
14
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45)
7
6
15
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45)
7
6
16
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36)
7
6
17
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45)
7
6
18
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45)
7
6
19
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36)
7
6
20
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45)
7
6
21
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36)
7
6
22
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27)
7
6
23
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
8Table 7
The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.
Lie bracket
Index
Nullity
1
(0, 0, 0, 0,  1.23,  1.24,  1.25  + 1.34,  1.17 +  1.26 +  1.35,  1.18,  1.27  + 1.36  + 1.45)
4
3
2
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
3
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.35,  1.18 +  1.27 +  1.45)
5
3
4
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.35,  1.18 +  1.27 +  1.36)
5
3
5
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.36 +  1.45)
5
3
6
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
7
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.36)
5
3
8
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.45)
5
3
9
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36)
5
3
10
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27)
5
3
11
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36 +  1.45)
5
3
12
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27 +  1.45)
5
3
13
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27 +  1.36)
5
3
14
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
15
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
5
3
16
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
5
3
17
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
18
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
19
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.45)
5
3
20
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36)
5
3
21
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27)
5
3
22
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
23
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
5
3
24
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
5
3
25
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
5
3
26
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
5
3
27
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
5
3
28
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
5
3
29
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
5
3
30
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
4
3
31
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
4
3
32
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
4
3
33
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.36)
4
3
34
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
4
3
35
(0, 0, 1.12,  1.13,  1.23,  0,  1.16 +  1.25,  1.17 +  1.26 +  1.35,  1.27 +  1.36 +  1.45)
4
3
36
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.45)
5
3
37
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.36)
5
3
38
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.27 +  1.36 +  1.45)
5
3
39
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.45)
5
3
40
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.36)
5
3
41
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36 +  1.45)
5
3
42
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27 +  1.45)
6
3
43
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27 +  1.36)
6
3
44
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
6
3
45
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
6
3
46
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
6
3
47
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
6
3
48
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
6
3
49
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.45)
5
3
50
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.36)
5
3
51
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.45)
5
3
52
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.36)
5
3
53
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.45)
5
3
54
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.36)
5
3
55
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27)
5
3
56
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.27 +  1.36 +  1.45)
5
3
57
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.45)
5
3
58
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.27 +  1.36 +  1.45)
5
3
59
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.45)
5
3
60
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
6
3
61
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
6
3
62
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
6
3
63
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
6
3
64
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
6
3
65
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
6
3
66
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.36 +  1.45)
6
3
67
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.45)
6
3
68
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.45)
6
3Table 8
The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.
Lie bracket
Index
Nullity
69
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.45)
6
3
70
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36)
6
3
71
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27)
6
3
72
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.45)
6
3
73
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18)
6
3
74
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.36 +  1.45)
6
3
75
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.45)
6
3
76
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.36)
6
3
77
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
6
3
78
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
6
3
79
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.27 +  1.36 +  1.45)
7
3
80
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.45)
7
3
81
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.36)
7
3
82
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.45)
7
3
83
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.36)
7
3
84
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.45)
7
3
85
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.36)
7
3
86
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27)
7
3
87
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.27 +  1.36 +  1.45)
6
3
88
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.45)
6
3
89
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.36)
6
3
90
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.45)
7
3
91
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36)
7
3
92
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27)
7
3
93
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18)
7
3
94
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.34,  1.17 +  1.26,  1.18)
7
3
95
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17,  1.18 +  1.45)
7
3
96
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17,  1.18 +  1.36)
7
3
97
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17 +  1.35,  1.18)
7
3
98
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25 +  1.34,  1.17,  1.18)
7
3


## 4.2.1. Candidates of Nilsoliton Metrics

Table5 illustrates how many possible candidates of Lie algebras appear in dimension 9 up to the nullity of its Gram matrix. The algebras illustrated in Table 4 are possible candidates of nilsoliton metric Lie algebras with ordered type of derivations in dimension 8. Here, as an example we give potential Lie algebra structures when the nullity of their corresponding Gram matrices are 3, 6 and 8 in Tables 6, 7, and 8 respectively for dimension nine.Table 4
The 8-dimensional nilsoliton metric Lie algebra candidates.
Lie bracket
Index
Nullity
1
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35)
6
4
2
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35)
6
4
3
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26)
6
4
4
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35)
6
5Table 5
Number of 9-dimensional nilsoliton metric Lie algebra candidates.
Nullity
Number of Lie algebras
3
98
4
81
5
45
6
22
7
7
8
1Table 6
The 9-dimensional candidates of nullities 6 and 8.
Lie bracket
Index
Nullity
1
(0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26  + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
6
6
2
(0, 0, 1.12, 1.13, 1.14, 1.15, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
3
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
4
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45)
7
6
5
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45)
7
6
6
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45)
7
6
7
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36)
7
6
8
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.36 + 1.45)
7
6
9
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
6
10
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45)
7
6
11
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45)
7
6
12
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36)
7
6
13
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45)
7
6
14
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45)
7
6
15
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45)
7
6
16
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36)
7
6
17
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45)
7
6
18
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45)
7
6
19
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36)
7
6
20
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45)
7
6
21
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36)
7
6
22
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27)
7
6
23
(0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45)
7
8Table 7
The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.
Lie bracket
Index
Nullity
1
(0, 0, 0, 0,  1.23,  1.24,  1.25  + 1.34,  1.17 +  1.26 +  1.35,  1.18,  1.27  + 1.36  + 1.45)
4
3
2
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
3
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.35,  1.18 +  1.27 +  1.45)
5
3
4
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.35,  1.18 +  1.27 +  1.36)
5
3
5
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.36 +  1.45)
5
3
6
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
7
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.36)
5
3
8
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.45)
5
3
9
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36)
5
3
10
(0, 0, 1.12,  0,  1.23,  1.24,  1.16  +  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27)
5
3
11
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36 +  1.45)
5
3
12
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27 +  1.45)
5
3
13
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27 +  1.36)
5
3
14
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
15
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
5
3
16
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
5
3
17
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
18
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.45)
5
3
19
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.45)
5
3
20
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26,  1.18 +  1.27 +  1.36)
5
3
21
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.25 +  1.34,  1.17  + 1.26 +  1.35,  1.18 +  1.27)
5
3
22
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
5
3
23
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
5
3
24
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
5
3
25
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
5
3
26
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
5
3
27
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
5
3
28
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
5
3
29
(0, 0, 1.12,  0,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
5
3
30
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
4
3
31
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
4
3
32
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
4
3
33
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.36)
4
3
34
(0, 0, 1.12,  0,  1.14 +  1.23,  1.24,  1.16 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
4
3
35
(0, 0, 1.12,  1.13,  1.23,  0,  1.16 +  1.25,  1.17 +  1.26 +  1.35,  1.27 +  1.36 +  1.45)
4
3
36
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.45)
5
3
37
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.36)
5
3
38
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.27 +  1.36 +  1.45)
5
3
39
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.45)
5
3
40
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.36)
5
3
41
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36 +  1.45)
5
3
42
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27 +  1.45)
6
3
43
(0, 0, 1.12,  1.13,  1.14,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27 +  1.36)
6
3
44
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.36 +  1.45)
6
3
45
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.45)
6
3
46
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
6
3
47
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
6
3
48
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
6
3
49
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.45)
5
3
50
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.36)
5
3
51
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.45)
5
3
52
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.36)
5
3
53
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.45)
5
3
54
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.36)
5
3
55
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27)
5
3
56
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.27 +  1.36 +  1.45)
5
3
57
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.27 +  1.45)
5
3
58
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.27 +  1.36 +  1.45)
5
3
59
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.45)
5
3
60
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
6
3
61
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.45)
6
3
62
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
6
3
63
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.45)
6
3
64
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.27 +  1.36)
6
3
65
(0, 0, 1.12,  1.13,  1.14,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18 +  1.27)
6
3
66
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.36 +  1.45)
6
3
67
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.45)
6
3
68
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.45)
6
3Table 8
The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.
Lie bracket
Index
Nullity
69
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.45)
6
3
70
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36)
6
3
71
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27)
6
3
72
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26,  1.18 +  1.45)
6
3
73
(0, 0, 1.12,  1.13,  1.14 +  1.23,  0,  1.16 +  1.25 +  1.34,  1.17 +  1.26 +  1.35,  1.18)
6
3
74
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.36 +  1.45)
6
3
75
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.45)
6
3
76
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27 +  1.36)
6
3
77
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36 +  1.45)
6
3
78
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27 +  1.36)
6
3
79
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.27 +  1.36 +  1.45)
7
3
80
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.45)
7
3
81
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.27 +  1.36)
7
3
82
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.45)
7
3
83
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.35,  1.18 +  1.27 +  1.36)
7
3
84
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.45)
7
3
85
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26,  1.18 +  1.27 +  1.36)
7
3
86
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.26 +  1.35,  1.18 +  1.27)
7
3
87
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.27 +  1.36 +  1.45)
6
3
88
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.45)
6
3
89
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.27 +  1.36)
6
3
90
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.45)
7
3
91
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17,  1.18 +  1.27 +  1.36)
7
3
92
(0, 0, 1.12,  1.13,  1.14,  1.15,  1.16 +  1.25 +  1.34,  1.17 +  1.35,  1.18 +  1.27)
7
3
93
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.34,  1.17 +  1.35,  1.18)
7
3
94
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.34,  1.17 +  1.26,  1.18)
7
3
95
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17,  1.18 +  1.45)
7
3
96
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17,  1.18 +  1.36)
7
3
97
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25,  1.17 +  1.35,  1.18)
7
3
98
(0, 0, 1.12,  1.13,  1.14 +  1.23,  1.15 +  1.24,  1.16 +  1.25 +  1.34,  1.17,  1.18)
7
3


## 5. Conclusion

In this work, we have focused on nilpotent metric Lie algebras of dimensions eight and nine with ordered type of derivations. We have given specific Jacobi identity conditions for Lie algebras which allowed us to simplify the Jacobi identity condition. We have classified nilsoliton metric Lie algebras for the corresponding Gram matrixU being invertible and noninvertible. For dimension 8, we have focused on nilsoliton metric Lie algebras with noninvertable Gram matrix which leads to more than one solution for Uv=[1]. We have proved that if the nilpotent Lie algebra admits a soliton metric with corresponding Gram matrix being noninvertible, all the solutions of Uv=[1] correspond to a unique derivation. This theorem has allowed us to omit several cases that come from nonordered eigenvalues without considering Jacobi condition. Moreover, we have classified some nilsoliton metric Lie algebras with derivation types 1<2<⋯<n and provided some candidates that may be classified. We are currently working on an algorithm that provides a full list of classifications for dimensions eight and nine.



---

*Source: 871930-2013-06-03.xml*