Derivation of a Lie algebra ($D[x,y]=[Dx,y]+[x,Dy]$)

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

Over a field of characteristic zero, a nilpotent operator $D$ which is a derivation of a Lie algebra exponentiates to a Lie algebra automorphism. The iterated rulefollows by induction using Pascal's identity. Regrouping the finite expansion of $[\exp (D)x,\exp (D)y]$ by total degree gives $\exp (D)[x,y]$. The inverse is $\exp (-D)$. This factorial construction must not be reduced naively to small positive characteristic.

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.

The derivations of a Lie algebra form a Lie subalgebra of $\mathrm{End}(\mathfrak{g})$ with the commutator bracket. The identity $[D,\mathrm{ad}x]=\mathrm{ad}(Dx)$ makes the inner derivations an ideal of a Lie algebra. For a complex semisimple Lie algebra, the Weyl complete reducibility theorem and the vanishing center of a Lie algebra imply $\mathrm{Der}(\mathfrak{g})=\mathrm{ad}\mathfrak{g}$.

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.