A Computational Approach for the Classifications of 8-dimensional
Nilsolitons

In this study, we present an algorithm in MATLAB to classify $8$
dimensional non-abelian nilsoliton metric Lie algebras with singular
Gram matrices. With this algorithm, we can compute Lie brackets,
structure constants, index, rank and eigenvalues of the nilsoliton
derivation. This paper can be considered as a follow up paper to our
previous study \cite{Kad3}. In our previous paper, we
defined an algorithm that helps to classify algebras that admit simple
derivations and singular Gram matrices $U$. Since the Gram matrices
are singular, there exists infinitely many solutions to
$Uv=[1]_m$, where the solutions are exactly the structure
constantsâ€™ squares. In order the algebra to be a Lie algebra, the
structure constants has to satisfy the Jacobi identity. In our previous
work, we did not present a methodology to classify algebras that satisfy
Jacobi identity. But in this paper, we extend the capability in such a
way that we are able to create and solve the Jacobi identity/identities
with the help of computer algorithms for each index set. Therefore we
completely classify all indecomposible nilsolitons in dimension $8$.
Several examples are provided for the illustration of the methodology.