= Zero generalized weight space of a Cartan subalgebra
{title2=$L^0(H)=H$}
Let $H$ be a <Cartan subalgebra> of a finite-dimensional complex <Lie algebra> $L$. In the adjoint <generalized-weight decomposition for a nilpotent Lie algebra>, its zero summand satisfies $L^0=H$. Nilpotence gives $H\subseteq L^0$. If $L^0/H$ were nonzero, all adjoint operators of $H$ on this quotient would be nilpotent, so the <Engel theorem> would supply a nonzero coset $x+H$ with $[H,x]\subseteq H$. That contradicts self-normalization. The derivation rule ensures that $L^0$ is a subalgebra, so the quotient action is well-defined.
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