= Semisimple Lie ideal has a commuting ideal complement
{title2=$L\trianglelefteq M\implies M=L\oplus L^{\perp_{\kappa_M}}$}
A finite-dimensional complex <semisimple Lie algebra> which is an ideal in a finite-dimensional <Lie algebra> has the displayed direct complement. The restriction of the <Killing form> of $M$ to $L$ equals the <Killing form> of $L$: the induced adjoint maps of elements of $L$ on $M/L$ vanish. This restriction is nondegenerate by the <Cartan criterion for semisimplicity>. Invariance makes its orthogonal complement an ideal. Nondegeneracy on $L$ gives a direct vector-space sum with that complement, and the bracket of the two ideals lies in their zero intersection, so they commute. The ambient algebra need not itself be semisimple.
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