My question is regarding the proof of Lemma 8.1c) in Kac's infinite-dimensional Lie algebras. I don't see how the claim follows from what is stated in the proof. I would be grateful for any clarification.
Preliminaries: Let $\psi$ be a finite-order automorphism of a simple, finite-dimensional Lie algebra $\mathfrak{g}$ of order $n$. We may then decompose $\mathfrak{g}$ into eigenspaces of $\psi$: $$\mathfrak{g} = \bigoplus_{j \in \mathbb{Z}_n}\mathfrak{g}_j .$$ Question: I want to show that $\mathfrak{g}_0$ is a reductive Lie algebra.
Proof sketch (from Kac): Let $\mathfrak{h}_0$ be the maximal ad-diagonalisable subalgebra of $\mathfrak{g}_0$ and let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra containing $\mathfrak{h}_0$. Consider the root decomposition $\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta} \mathfrak{g}_{\alpha}$ with respect to this Cartan subalgebra. Take the root subspace $\Delta' = \{\alpha \in \Delta : \alpha|_{\mathfrak{h}_0}=0 \}$. The centraliser $C_{\mathfrak{g}}(\mathfrak{h}_0)$ of $\mathfrak{h}_0$ in $\mathfrak{g}$ then takes the form $$C_{\mathfrak{g}}(\mathfrak{h}_0) = \mathfrak{h} + \sum_{\alpha \in \Delta'}\mathfrak{g}_{\alpha}.$$ From this he concludes that $\mathfrak{g}_0$ is reductive, since $C_{\mathfrak{g}}(\mathfrak{h}_0) = \mathfrak{h}+\mathfrak{s}$ for a semisimple Lie algebra $\mathfrak{s}$ with trivial intersection with $\mathfrak{g}_0$.
Problem: I do not understand the last steps from the proof and thus do not understand the conclusion.