 Cartan solvability criterion

A complex Lie subalgebra $\mathfrak{h}\subseteq \mathfrak{gl}(V)$ is solvable iffor every $x\in [\mathfrak{h},\mathfrak{h}]$ and $y\in \mathfrak{h}$. For an abstract Lie algebra, this is equivalent to $\kappa (\mathfrak{h},[\mathfrak{h},\mathfrak{h}])=0$.