Generalized weight space of a Lie algebra representation ($V^{\lambda }=\{v:(\rho (h)-\lambda (h)I{)}^{\dim V}v=0\forall h\}$)

For a finite-dimensional complex Lie algebra representation $\rho$ of a nilpotent Lie algebra $H$, a generalized weight space is the simultaneous generalized eigenspace displayed above, where $\lambda :H\to \mathbb{C}$ is a linear functional. Every occurring $\lambda$ vanishes on the derived algebra of $H$. Ordinary weight vectors satisfy the same equations without the powers.

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 $(\mathrm{ad}h{)}^{N}k=0$, hence $(\mathrm{ad}\rho (h){)}^N\rho (k)=0$. 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.

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.