Question: Let $L$ be an reductive lie algebra,$H$ is an ideal of $L$.Let $S$ be the centralizer of $H$ in $L$, If $S$ is abelian, proof that $\left[ L,L\right] \subseteq H$.
My attempt:Since $S$ be the centralizer of $H$ in $L$ and abelian, so it is solvable, it is easy to see that $Z\left( L\right) \subseteq S$,but $L$ reductive, so $Rad\left( L\right) =Z\left( L\right) $,hence $Z\left( L\right) =S$.I don't know where to go from here.