= Self-centralizing property of a Cartan subalgebra
{title2=$C_L(H)=H$}
For a <Cartan subalgebra> $H$ of a finite-dimensional complex <semisimple Lie algebra> $L$, $H$ is abelian and $C_L(H)=H$, so it is maximal among abelian subalgebras. Starting from the nilpotent self-normalizing definition, the <generalized-weight decomposition for a nilpotent Lie algebra> and the <Engel theorem> show that the zero generalized <weight space> of its adjoint action is exactly $H$. Invariance of the <Killing form> makes $H$ orthogonal to every nonzero generalized <weight space>. The <Lie theorem> gives $B([H,H],H)=0$, and nondegeneracy then forces $[H,H]=0$. An element commuting with $H$ normalizes it, hence belongs to $H$.
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