Derivation of a Lie algebra

Summary

A derivation of a Lie algebra $\mathfrak g$ is a linear map $D$ on $\mathfrak g$ that obeys the product rule $D[x, y] = [Dx, y] + [x, Dy]$. It is the special case of a derivation of an algebra when the product is the Lie bracket. The derivations of $\mathfrak g$ form a Lie algebra $\operatorname{Der}(\mathfrak g)$ under the commutator. The maps $\operatorname{ad}_x = [x, \cdot]$ are derivations, called inner derivations. For a finite-dimensional semisimple Lie algebra over a field of characteristic $0$, every derivation is inner.