= Orthogonal ideal splitting for a nondegenerate Killing form
{title2=$\mathfrak g=I\oplus I^\perp$}
For an <ideal of a Lie algebra> $I$ and nondegenerate <Killing form>, invariance makes $I^\perp$ an ideal. An element of $I\cap I^\perp$ commutes with $I$, so the intersection is an abelian ideal. <Abelian ideals lie in the radical of the Killing form>, forcing the intersection to vanish. Thus $\mathfrak g=I\oplus I^\perp$ as commuting ideals. Iterating minimal nonzero ideals gives the direct-sum formulation of a <semisimple Lie algebra>.
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