Inner derivation of a Lie algebra ($\mathrm{ad}\mathfrak{g}\subseteq \mathrm{Der}(\mathfrak{g})$)

An inner derivation of a Lie algebra is ${\mathrm{ad}}_x:y\mapsto [x,y]$. The Jacobi identity makes this a derivation of a Lie algebra. For every derivation of a Lie algebra $D$, $[D,{\mathrm{ad}}_x]={\mathrm{ad}}_{Dx}$, so the inner derivations form an ideal of a Lie algebra in the derivation Lie algebra. The kernel of $x\mapsto {\mathrm{ad}}_x$ is the center of a Lie algebra.

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 $x=0$.