# Symplectic analog of Lie--Poisson bracket

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 $$\omega = f_{ab}^c z^c d z^a \wedge d z^b$$)

For $$z_i$$ a basis of the Lie algebra $$\mathfrak{g}$$ and $$f_{ab}^c$$ its structure constant, $$\omega= \sum_{\substack{a should define a closed two form on the dual of $$\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.