Theorem 3.3. of Humphreys goes something like this: given a subalgebra $\mathfrak{g}$ of $\mathfrak{gl}(V)$ where $V$ is nonzero, finite-dimensional and $\mathfrak g$ consists of nilpotent endomorphisms, then there is a nonzero vector $v \in V$ such that ${\frak g}.v = 0$. Assuming I understand the first part of the proof I understand the rest, however I am not confident that is the case. I paraphrase the first part of the proof:
By induction on $\dim \frak g.$Suppose $\frak h \subset \frak g$ a proper subalgebra. $\frak h$ acts via ad as a Lie algebra of nilpotent linear transformations on the vector space $\frak g$, hence on $\frak g / \frak h$. As the dimension of $\frak h$ is strictly smaller than that of $\frak g$, the induction hypothesis ensures a nonzero $x + \frak h$ killed by the image of $\frak h$ in ${\frak gl}(\frak g /\frak h)$.
Breaking this down, I want to confirm that this is the situation: we have a composite $$\frak h \to {\frak gl}(\frak g) \to {\frak gl}(\frak g / \frak h)$$ the first map being $y \mapsto \text{ad}_y(-)$, the latter being the canonical projection. For dimensional reasons the theorem is in this particular situation assumed true (by the inductive hypothesis), and nilpotency of $\text{ad}_y(-)$ follows from nilpotency of $y$, so we are in the situation of the theorem. Here I am starting to lose confidence; where is the subspace of $\frak gl(\frak g / \frak h)$? Why is the theorem valid for this composite?
The induction hypothesis is applied for the vector space $V=\mathfrak{g}/\mathfrak{h}$, which is of lower dimension, since $\mathfrak{h}$ is a proper subalgebra. So you find a vector $x+\mathfrak{h}\neq \mathfrak{h}$ in $V$ killed by the image of $\mathfrak{h}$ in $\mathfrak{gl}(V)$. It follows that $\mathfrak{h}$ is properly contained in its normalizer $N_{\mathfrak{g}}(\mathfrak{h})$, and the proof can go on. I do not see why you need composite maps.