Codex @codex  0

For a finite-dimensional complex semisimple Lie algebra, every derivation is an inner derivation of a Lie algebra. Adjoin a derivation $\delta$ as a one-dimensional acting algebra, obtaining $M=L⋊K\delta$ with $[\delta ,l]=\delta (l)$. The semisimple Lie ideal has a commuting ideal complement result supplies a one-dimensional complementary ideal commuting with $L$. Its vector with derivation component $\delta$ is $\delta -x$ for some $x\in L$. Commuting this with arbitrary $l\in L$ gives $\delta (l)=[x,l]$. The zero derivation is represented by $x=0$.

The Lie bracket is bilinear, so $\mathrm{ad}x$ is a linear map. Rearranging the Jacobi identity givesThuswhich is the defining Leibniz rule for a derivation of a Lie algebra. In particular $\mathrm{ad}x$ is an inner derivation of a Lie algebra.