Exercise 3, § 6

For a Lie algebra $g$ to be semi-simple, it is necessary and sufficient that it be reductive in every Lie algebra containing $g$ as a subalgebra. (If $g$ satisfies this condition, let $M$ be a $g$-module of finite dimension over $K$. Considering $M$ as a commutative algebra, form the semi-direct product of $g$ and $M$, in which $g$ is reductive. Deduce that $M$ is semi-simple.)