Any coadjoint orbit $\mathscr O \subset \mathfrak g^∗$ has a symplectic structure $\omega_{\mathscr O}$ which satisfies:
$\forall \xi \in \mathscr O$; $v, w \in \mathfrak g$: $\omega_{\mathscr O}((ad^∗_v)_\xi, (ad^∗_w)_\xi) = −\xi([v, w])$,
How do I proof that?
I know the following definitions:
$G$ :Lie group with Lie algebra $\mathfrak g$. The coadjoint action of $G$ on $\mathfrak g^∗$ is
$Ad^∗ : G \times \mathfrak g^∗ \rightarrow \mathfrak g^∗, (Ad^∗_g\xi)(v) := \xi(Ad_{g^{−1}}(v))$,
and the induced infnitesimal action is:
$ad^∗ : \mathfrak \rightarrow \mathfrak X(\mathfrak g^∗), (ad^∗ v)\xi := \frac {d}{dt}|_{t=0} Ad^∗ \exp(−tv)(\xi)$
How do I continue from here?