= Killing radical is a solvable ideal
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{title2=$\operatorname{rad}B_L\subseteq\operatorname{rad}L$}
For a finite-dimensional complex <Lie algebra> $L$, the radical $R$ of its <Killing form> is a <Lie algebra ideal> by invariance. For $x,y\in R$, their adjoint actions vanish on $L/R$, so $B_R(x,y)=B_L(x,y)=0$. The <Cartan solvability criterion> makes the image $\operatorname{ad}_R(R)$ solvable; its kernel is the abelian center, so $R$ is solvable. A <semisimple Lie algebra> has no nonzero solvable ideal, forcing its Killing radical to vanish.
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