Cartan's Criterion - Cartan's Criterion For Semisimplicity

Cartan's Criterion For Semisimplicity

Cartan's criterion for semisimplicity states:

A finite-dimensional Lie algebra over a field of characteristic zero is semisimple if and only if the Killing form is non-degenerate.

Dieudonné (1953) gave a very short proof that if a finite dimensional Lie algebra (in any characteristic) has a non-degenerate invariant bilinear form and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.

Conversely, it follows easily from Cartan's criterion for solvability that a semisimple algebra (in characteristic 0) has a non-degenerate Killing form.