Say I have a Lie algebra, $\mathfrak{g}$, and some element of the dual $m \in \mathfrak{g}^*$ with the property that $m([u,v])=0$ for all $u,v \in \mathfrak{g}$ ($m$ non-zero). What can be said about the Lie algebra $\mathfrak{g}$? For one it cannot be semi-simple. It is true when $\mathfrak{g}$ is the semi direct product of a one dimensional Lie algebra and $m$ in the corressponding dual vector space (I hope my terminology makes sense). Are there other cases or is a Lie algebra fulfilling these conditions necessarily a semi direct product of a one dimensional Lie algebra with something else?
Unfortunately I don't know enough about Lie groups to know if this is more appropriate for overflow or stack exchange, if I should delete and repost on stack exchange please let me know.