= Derivations of a semisimple Lie algebra are inner
{title2=$\operatorname{Der}(L)=\operatorname{ad}L$}
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 $M=L\rtimes K\delta$ 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 $x=0$.
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