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 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. Lie's theorem