Uniqueness of Levi decomposition

Question

Let $g$ be a Lie algebra. It is a well-known fact that it can be written as $g=r \oplus s$ for the radical $r$ and a semisimple Lie algebra $s$. I would like to know whether this decomposition is unique, i.e. whether $s$ is uniquely determined.

Answer 1

The subalgebra $s$ is unique up to an automorphism of the form exp(ad$z$) where ad$z$ is the inner derivation of $g$ determined by an element $z$ of the nilradical (the largest nilpotent ideal of $g$). This was proved by Malcev in A.I. Malcev, "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra" Dokl. Akad. Nauk SSSR , {\bf 36} : 2 (1942) pp. 42–45 (In Russian).