Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 1 / 1 / Solution

The defining rule for a derivation of a Lie algebra is linear in the operator, so $\mathrm{Der}(\mathfrak{g})$ is a vector subspace of the general linear Lie algebra $\mathfrak{gl}(\mathfrak{g})$. Its Lie bracket is the operator commutator. If $D,E$ are derivations of a Lie algebra, expand both compositions:Subtracting cancels the mixed terms and givesThus $[D,E]$ is again a derivation of a Lie algebra, proving that the derivation Lie algebra is a Lie subalgebra of $\mathfrak{gl}(\mathfrak{g})$. This calculation works in every characteristic of a field.

The Jacobi identity states $[x,[y,z]]=[[x,y],z]+[y,[x,z]]$. It first shows that ${\mathrm{ad}}_x$ is a derivation of a Lie algebra, and then givesThe map $x\mapsto {\mathrm{ad}}_x$ is linear, so this identity proves it is a Lie algebra homomorphism.

Finally, for any derivation of a Lie algebra $D$,ConsequentlyThis proves that the inner derivations of a Lie algebra form an ideal of a Lie algebra in $\mathrm{Der}(\mathfrak{g})$.