defined by the formula where tr denotes the trace of a linear operator.
The converse can be deduced from the nilpotency criterion based on the Jordan–Chevalley decomposition, as explained there.
Applying Cartan's criterion to the adjoint representation gives: Cartan's criterion for semisimplicity states: Jean 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.
The converse is false: there are non-nilpotent Lie algebras whose Killing form vanishes.