With every finite-dimensional Lie algebra $\mathfrak{g}$ one can associate a linear Poisson bracket on $\mathfrak{g}^\ast$. With some more restrictions on $\mathfrak{g}$ and some extra ingredients, there are other constructions of linear, quadratic and cubic Poisson brackets associated with $\mathfrak{g}$: see Nonlinear Poisson structures and $r$-matrices by Li–Parmentier (1989) or §10.3 in the book Poisson Structures by Vanhaecke et al (2013).
The quadratic and cubic constructions assume in particular that $\mathfrak{g}$ is the Lie algebra of an associative algebra, and that $\mathfrak{g}$ is equipped with a non-degenerate symmetric bi-linear map $\langle \cdot | \cdot \rangle$ which satisfies $\langle xy | z\rangle = \langle x | yz\rangle$ for all $x,y,z \in \mathfrak{g}$. For example, $\mathfrak{g} = \mathfrak{gl}_n(\mathbb{R})$ and $\langle x | y \rangle = \operatorname{tr}(xy)$.
Now, e.g., for the Lie algebra of strictly upper-triangular matrices, the trace form is zero identically which is very degenerate, so the aforementioned quadratic and cubic constructions do not work.
Question: Is there a construction of a nonlinear Poisson bracket associated with a nilpotent finite-dimensional Lie algebra $\mathfrak{g}$?
I doubt there is such a construction generalizing the previously mentioned ones, so I rather ask if there is any (different) construction specific (or applicable) to the nilpotent case.
