Let $\mathfrak{g}$ be a Lie algebra. Let $C \subset \mathfrak{g}^*$ be the subspace of linear forms that vanish on Lie brackets: $$C = \{\alpha \in \mathfrak{g}^*, ~ [\mathfrak{g}, \mathfrak{g}] \subset \ker \alpha\}$$
If I'm not mistaken, $C$ can also be described as the fixed points of the coadjoint action, or as the first cohomology space $H^1(\mathfrak{g}, \mathbb{R})$.
Apparently, some people also call it the "center" of $\mathfrak{g}^*$. Why?