# Softwares to determine semi-simple types of Lie algebras generated over \$mathbb{R}\$ or \$mathbb{C}\$ by a set of matrices

I wish to determine the type of a Lie algebra generated over $$mathbb{R}$$ or $$mathbb{C}$$ by a set of square matrices with irrational elements. For example,

begin{align} n^+ = begin{pmatrix} 0 & -frac{1}{sqrt{2}} & 0 & 1 \ 0 & 0 & -frac{1}{sqrt{2}} & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & frac{1}{sqrt{3}} & 0 end{pmatrix}, quad n^- = begin{pmatrix} 0 & 0 & 0 & 0 \ frac{1}{sqrt{2}} & 0 & 0 & 0 \ 0 & frac{1}{sqrt{2}} & 0 & 1 \ frac{1}{sqrt{3}} & 0 & 0 & 0 end{pmatrix}, quad n^3 = begin{pmatrix} frac{1}{sqrt{2}} & 0 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & -frac{1}{sqrt{2}} & 0 \ 0 & -frac{1}{sqrt{3}} & 0 & 0 end{pmatrix}. end{align}
I’ve been using `GAP` and it is quite good at determining the type of Lie algebras over $$mathbb{Q}$$, which seems to work fine when all the matrix elements are rational numbers. However, when the matrices contain irrational numbers, if I still work with Lie algebras over $$mathbb{Q}$$, I would get a larger Lie algebra, and in the above example, I got 8 copies of $$A_1$$. On the other hand, If I work with Lie algebras over cyclotomic numbers, the function `SemiSimpleType` would break down. Therefore I wish to know:

Is there a way of doing this with `GAP`? If not, are there any other softwares which can do this?

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