I am struggling with the following logic which appears in Humphreys' introduction to semisimple Lie algebras.
Suppose that $\mathfrak{h}$ is a Cartan subalgebra in a semisimple $\mathfrak{g}$. Since $\mathfrak{h}$ is abelian (because it's toral), we have that $Ad_{\mathfrak{h}}$ is a commuting family of semisimple endomorphisms of $\mathfrak{g}$. This means that the family $Ad_{\mathfrak{h}}$ is simultaneously diagonalizable.
It follows, by inducting on the dimension of $\mathfrak{g}$, that $\mathfrak{g}$ is a direct sum of the subspaces $\mathfrak{g}^{\alpha} = \{x \in \mathfrak{g} | [a,x] = \alpha(a)x \; \forall a \in \mathfrak{h}\}$, where $\alpha$ ranges over $\mathfrak{h}^*$.
I understand the first paragraph. I don't understand how inducting on the dimension of $\mathfrak{g}$ gives us that $\mathfrak{g}$ is the direct sum as suggested. I think it's a result from linear algebra, but I can't find a proof of it.