Is decomposition of a semisimple Lie algebra unique?

Question

A semisimple Lie algebra is defined to be the sum of simple Lie algebras. But is this decomposition to simple Lie algebras unique? If not can you give an example?

Answer 1

According to Wikipedia, a Lie algebra is called semisimple if it is a direct sum of simple Lie algebras (and a Lie algebra is called simple if its adjoint representation is irreducible, i.e., if it has no proper ideal). The decomposition $L=L_1\oplus \cdots \oplus L_n$ is unique up to permutations of the summands. Indeed, the simple ideals $L_i$ are uniquely determined by $L$, and we have $[L_i,L_j]\subseteq I_i\cap I_j=0$.