I am currently following the MIT lie algebra notes (for fun, not for h.w.!) and I am stuck on a step in one of the main theorems of the lecture. The notes are found here, https://math.mit.edu/classes/18.745/classnotes.html , and the question is with regards to theorem 6.6 of lecture 6. The theorem is as follows:
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and π its representation on a finitedimensional vector space V , over an algebraically closed field $\mathbb{F}$ of characteristic 0. Let $\mathfrak{h}$ be a nilpotent subalgebra of $\mathfrak{g}$. Then the following equalities hold.
$$V = \bigoplus_{\lambda \in \mathfrak{h}*}V^{\mathfrak{h}}_{\lambda} \ \ \ (2)$$
$$ \pi(\mathfrak{g}^{\mathfrak{h}}_{\alpha}) \ V^{\mathfrak{h}}_{\lambda} \subset V^{\mathfrak{h}}_{\lambda+ \alpha} \ \ \ (3)$$
where $V^{\mathfrak{h}}_{\lambda}$ and $\mathfrak{g}^{\mathfrak{h}}_{\alpha}$ are the generalised weight spaces w.r.t to the rep $\pi$ and the adjoint rep.
The proof of this theorem is given below:
I am stuck on the establishment of (2), I don't get what induction is being done on. Any help would be greatly appreciated!!