= Abelian ideals lie in the radical of the Killing form
If $\mathfrak a$ is an abelian <ideal of a Lie algebra>, then $\operatorname{ad}_x\operatorname{ad}_y$ maps $\mathfrak g$ into $\mathfrak a$ and vanishes on $\mathfrak a$ whenever $x\in\mathfrak a$. Its trace is zero, so $\kappa(\mathfrak a,\mathfrak g)=0$. Every nonzero solvable ideal has a last nonzero term in its <derived series of a Lie algebra>, which is such an abelian ideal. Consequently a nondegenerate <Killing form> forces the <solvable radical> to vanish.
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