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?