Maximal reductive subalgebra of $\mathfrak{g}^e$.

Question

I cannot figure out how to prove the following statement, please help:

Let $e$ be a non-zero nilpotent element of $g$ and let $\mathfrak{s} =\{e, h, f\}$ be an $sl_2$-triple.

Then $\mathfrak{g}^\mathfrak{s}$ (the centralizer of $\mathfrak{s}$ in $\mathfrak{g}$) is a maximal reductive subalgebra of $\mathfrak{g}^e$.