I'm starting to learn about Lie algebras in order to decompose a complex 16-dimensional Lie algebra that I have given in terms of the whole multiplication table for the Lie bracket. I want to decompose it into a sum of simple Lie algebras. I should be able to decompose it according to the Levi decomposition and a sum of simple Lie algebras?
Could you please outline the steps and techniques that I will need to do this decomposition? It may help me select the material I need to learn.
Is there a computer package to do that?
First check the Killing form of $L$. If it is non-degenerate, then $L$ is semisimple, so that $$ L=L_1\oplus \cdots \oplus L_n, $$ with simple ideals $L_i$. Because of $\dim(L)=16$ we only can have $\dim(L_i)\in \{3,8,10\}$, and here we only have the possibilities $16=8+8=10+3+3$, i.e., $$ L\cong \mathfrak{sl}_3(\Bbb C)\oplus \mathfrak{sl}_3(\Bbb C),\quad L\cong \mathfrak{so}_5(\Bbb C)\oplus \mathfrak{sl}_2(\Bbb C) \oplus \mathfrak{sl}_2(\Bbb C). $$
If not, apply the Levi decomposition $L=S\rtimes {\rm rad}(L)$ using GAP and repeat the above decomposition with the Levi subalgebra. The solvable radical ${\rm rad}(L)$ then is left.