Corollary 3

If $g$ is semi-simple, every derivation of $g$ is inner.

$\operatorname{ad} g$ is isomorphic to $g$ and hence semi-simple and is an ideal of the Lie algebra $\delta$ of derivations of $g$ ($§ 1$, Proposition 1). If $D \in \delta$ commutes with the elements of $\operatorname{ad} g$, then, for all $x \in g$, $\operatorname{ad} D(x) = [D, \operatorname{ad} x] = 0$, whence $D(x) = 0$; hence $D = 0$. Corollary 3 then follows from Corollary 1.