Zero generalized weight space of a Cartan subalgebra ($L^0(H)=H$)

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.