The coisotropic orbit characterization for symplectic representations

Let $({\mathfrak m},V,\omega)$ be a symplectic representation of the Lie algebra ${\mathfrak m}$. A subspace $W\subseteq V$ is coisotropic when $W^{\perp_\omega}\subseteq W$, where $W^{\perp_\omega}=\{v\in V:\omega(v,w)=0\text{ for all }w\in W\}$. An ${\mathfrak m}$-equivariant symmetric bilinear map is a symmetric bilinear map compatible with the ${\mathfrak m}$-actions on $V$ and ${\mathfrak m}$. Coisotropic orbit characterization. The subspace ${\mathfrak m}\cdot A$ is coisotropic for all nonzero $A\in V$ if and only if there exists an ${\mathfrak m}$-equivariant symmetric bilinear map

$$B_{\mu}:V\times V\rightarrow {\mathfrak m}$$

which satisfies the identity referred to in the source as. This would characterize symplectic representations whose nonzero infinitesimal orbits are all coisotropic by the existence of the additional bilinear map. The preceding theorem proves that the coisotropy condition holds for special symplectic representations, but the converse characterization is presented as a conjecture and its status is not resolved in the supplied text.