If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, where can I find a proof?)

It is for every ideal $ \mathfrak{a}$ of $\mathfrak{g}$ there exists an ideal $ \mathfrak{b}$ sucht that $\mathfrak{g}=\mathfrak{a}+\mathfrak{b}$ or equivalently the adjoint representation is semi-simple