As remarked by Jim Humphreys in a comment to my answer to a previous question, the notion of *reductive* for a Lie algebra (in characteristic zero) has no intrinsic interest, which means that the answer to this question has to be positive.

Here is one possible construction. Let $\mathfrak{g} = \mathfrak{s} \oplus \mathfrak{z}$ be a reductive Lie algebra, where $\mathfrak{s}$ is semisimple and $\mathfrak{z}$ is the centre of $\mathfrak{g}$. Consider a representation $V$ of $\mathfrak{g}$ where $\mathfrak{z}$ does *not* act semisimply. Now let $\mathfrak{h}$ be the semidirect product $\mathfrak{g} \ltimes V$, with $V$ abelian.