= Generalized-weight decomposition for a nilpotent Lie algebra
{title2=$V=\bigoplus_\lambda V^\lambda$}
Every finite-dimensional complex <Lie algebra representation> of a <nilpotent Lie algebra> decomposes into invariant <generalized weight spaces>. The nilpotence assumption is essential: a general solvable action can have extensions between distinct weights.
Here is a proof. For $h,k\in H$, nilpotence gives $(\operatorname{ad}h)^N k=0$, hence $(\operatorname{ad}\rho(h))^N\rho(k)=0$. By the <nilpotent commutator preserves generalized eigenspaces> lemma, every <generalized eigenspace> of $\rho(h)$ is invariant under every $\rho(k)$. Split successively by the acting operators of a <basis> of $H$. Each resulting block is invariant under $H$, and each basis operator has only one <eigenvalue> on it. The <Lie theorem> triangularizes the action on each block; its diagonal entries therefore give one and the same <linear functional> $\lambda$ on every basis vector of that block. Every $\rho(h)-\lambda(h)I$ is strictly upper triangular there, proving that the block is $V^\lambda$. Distinct blocks have distinct weights, so this is a <direct sum>. On the zero generalized <weight space>, every acting operator is a <nilpotent endomorphism>, so the <Engel theorem> supplies a nonzero common annihilated vector whenever that space is nonzero.
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