Cartan’s criterion for semisimplicity - Imported Knowlpedia Content

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

B(X,Y)=\mathrm{tr}(\mathrm{ad}_X\circ \mathrm{ad}_Y).

Theorem (Cartan). The following are equivalent:

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

Remarks

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.