Cartan's first criterion

Statement
Let $$F$$ be an algebraically closed field of characteristic zero. Let $$L$$ be a finite-dimensional Lie algebra over $$F$$. Let $$\kappa$$ denote the fact about::Killing form on $$L$$. Then, the following two statements are equivalent:


 * 1) $$L$$ is a solvable Lie algebra.
 * 2) $$\! \kappa(x,y) = 0$$ for all $$x \in L$$ and $$y \in [L,L]$$.

Related facts

 * Engel's theorem
 * Lie's theorem
 * Cartan's second criterion

Breakdown for other fields

 * Cartan's first criterion fails for non-algebraically closed fields
 * Cartan's first criterion fails for prime characteristic

Facts used

 * 1) uses::Lie's theorem