Derivation of a Lie algebra

A derivation of a Lie algebra is a linear map $D:L\to L$ satisfyingEvery adjoint map $\mathrm{ad}z:x\mapsto [z,x]$ is a derivation.

Every nonzero finite-dimensional nilpotent Lie algebra over a field of characteristic zero has a derivation outside its adjoint image. This remains true for characteristically nilpotent Lie algebras: all their derivations may be nilpotent, but they cannot all be inner.

If $D$ is a derivation of a Lie algebra, then its generalized eigenspaces satisfyApply a sufficiently large power of $D-(\lambda +\mu )I$ and use the binomial theorem together with the derivation rule.