Symplectic analog of Lie--Poisson bracket

Question

Given a Lie algebra one can construct a Poisson manifold using the Lie--Poisson bracket ($\{ z_{a},z_b \}=f_{ab}^c z_c$ where the $f$ are structure constants of a Lie algebra). It is well known that one needs some work to obtain the symplectic structures on the coadjoint orbits.

My question is if there exists a presymplectic analog of the Lie--Poisson bracket? By that I mean closed, two-form that is not necessarily nondegenerate, that one can also generically and directly define.

(E.g., something in the spirit of $\omega = f_{ab}^c z^c d z^a \wedge d z^b$)

Answer 1

For $z_i $ a basis of the Lie algebra $\mathfrak{g}$ and $f_{ab}^c$ its structure constant, $\omega= \sum_{\substack{a<b\\ d}}f_{ab}^cf_{cd}^ez_d \mathrm{d} z_a\wedge \mathrm{d}z_b$ should define a closed two form on the dual of $\mathfrak{g}$ for any $e$ by the Jacobi identity, if I am not mistaken. However, I have no idea if and where it appears in the literature.