Killing form nondegeneracy criterion - Imported Knowlpedia Content

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$ , and let $B$ be its Killing form.

Theorem (Cartan criterion via Killing form). $\mathfrak g$ is semisimple if and only if the Killing form $B$ is nondegenerate.

Remarks

Comments on the proof.

If $\mathfrak g$ is semisimple, then the adjoint representation is faithful modulo the center (see ker(ad)=center and trivial center for simple algebras), and complete reducibility arguments show that the invariant form $B$ must be nondegenerate.
Conversely, if $B$ is nondegenerate, then any solvable ideal forces degeneracy: the radical (maximal solvable ideal) lies in the radical of $B$ , so nondegeneracy implies the radical is zero, hence $\mathfrak g$ is semisimple.

Context. This theorem is often presented alongside Cartan’s criterion and is a key structural tool in the study of semisimple algebras and their representations.