Corollary 1

Let $g$ be a Lie algebra, $\beta$ its Killing form and $a$ a semi-simple sub-algebra of $g$. The orthogonal $h$ of $a$ with respect to $\beta$ is a supplementary subspace of $a$ in $g$ and $[a, h] \subset h$. If $a$ is an ideal of $g$, so is $h$, which is then the centralizer of $a$ in $g$.

Let $\beta'$ be the restriction of $\beta$ to $a$: it is the bilinear form associated with the representation $x \mapsto \operatorname{ad}_g x$ of $a$ on the space $g$. This representation is faithful and hence $\beta'$ is non-degenerate (Proposition 1). Hence $h$ is supplementary to $a$ in $g$. On the other hand, if $x, y$ are in $a$ and $z \in h$, then

since $[x, y] \in a$, and hence $[y, z] \in h$, which proves that $[a, h] \subset h$. If $a$ is an ideal of $g$, we know that $h$ is an ideal of $g$ ($§ 3$, Proposition 7) and $g$ is identified with $a \times h$. As the centre of $a$ is zero, the centralizer of $a$ in $g$ is $h$.