Let $\mathfrak{g}$ be a Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi$. Show that $C_\mathfrak{g}(h)$ is reductive, that is $Z(C_\mathfrak{g}(h))=Rad(C_\mathfrak{g}(h))$, for all $h\in\mathfrak{h}$.
For brevity, put $C=C_\mathfrak{g}(h)$, $Z=Z(C)$, and $R=Rad(C)$. Obviously $Z$ is a solvable ideal in $C$, so I just need to show it's maximal. Is there a "nice" way of showing maximality?