Killing form nondegeneracy criterion

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

The proof uses Cartan’s criterion to relate the solvable radical of $\mathfrak g$ to traces in the adjoint representation. Thus nondegeneracy is not merely a consequence of the adjoint representation being faithful; it detects the absence of nonzero solvable ideals.