Lemma 1

Let $g$ be a semi-simple Lie algebra. The adjoint representation of $g$ is semi-simple. Every ideal and every quotient algebra of $g$ is semi-simple.

Let $a$ be an ideal of $g$. The orthogonal $b$ of $a$ in $g$ with respect to the Killing form is an ideal of $g$ and $a \cap b$ is a commutative ideal ($§ 3$, no. 6, Proposition 7) and hence zero. Hence $b$ is supplementary to $a$ in $g$. Moreover, as the Killing form of $g$ is non-degenerate, so are its restrictions to $a$ and $b$ (Algebra, Chapter IX, $§ 4$, no. 1, Corollary to Proposition 1) and hence $a$ and $b$ are semi-simple (no. 1, Theorem 1 and $§ 3$, no. 6, Proposition 9).