All Derivations of Semisimple Lie Algebras are Inner

It turns out that all the derivations on a semisimple Lie algebra are inner derivations. That is, they’re all of the form for some . We know that the homomorphism is injective when is semisimple. Indeed, its kernel is exactly the center , which we know is trivial. We are asserting that it is also surjective, and thus an isomorphism of Lie algebras.

If we set and , we can see that . Indeed, if is any derivation and , then we can check that

This makes an ideal, so the Killing form of is the restriction of of the Killing form of . Then we can define to be the subspace orthogonal (with respect to ) to , and the fact that the Killing form is nondegenerate tells us that , and thus .

Now, if is an outer derivation — one not in — we can assume that it is orthogonal to , since otherwise we just have to use to project onto and subtract off that much to get another outer derivation that is orthogonal. But then we find that

since this bracket is contained in . But the fact that is injective means that for all , and thus . We conclude that and that , and thus that is onto, as asserted.