Weight spaces of Lie algebra modules

Abstract:

Let $V$ be a finite-dimensional module for the finite-dimensional Lie algebra $L$ over a field of characteristic zero. If ${V^\lambda } = \{ v \in V|\;{\text {all }}x \in L,{[x - \lambda (x)]^i}v = 0\;{\text {for some }}\}$ is nonzero, then $\lambda \in {L^*}$ and is a character of $L$. Moreover, the corresponding eigenspace $\{ v \in V|{\text {all }}x \in L,xv = \lambda (x)v\}$ is nonzero and ${V^\lambda }$ is an $L$ submodule of $V$.