Cartan’s criterion for semisimplicity

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic $0$. Let $B$ denote the Killing form on $\mathfrak{g}$, defined by

Theorem (Cartan). The following are equivalent:

1. $\mathfrak{g}$ is semisimple .
2. The Killing form $B$ is nondegenerate on $\mathfrak{g}$, i.e. $B(X,Y)=0$ for all $Y$ implies $X=0$.

Context. This criterion is a practical test for semisimplicity: it converts the intrinsic condition “$\mathfrak{g}$ has no nonzero solvable ideals” into a bilinear-algebra statement. It is complementary to Cartan’s criterion for solvability , which detects when a Lie algebra is solvable via vanishing of certain traces.

Remark. Nondegeneracy of $B$ implies strong structure results, including the decomposition of any semisimple Lie algebra into a direct sum of simple ideals .