 Closure of the inverse of a nondegenerate Poisson bivector

A nondegenerate Poisson bivector inverts to a closed differential form of degree two. Differentiating $S=P^{-1}$ gives ${\partial }_{d}S=-S({\partial }_{d}P)S$. Contracting the coordinate Jacobi condition for a Poisson bivector with three copies of $S$ yields ${\partial }_i{S}_{jk}+{\partial }_j{S}_{ki}+{\partial }_k{S}_{ij}=0$, which is precisely closure of the resulting symplectic form. An overall sign in the inverse convention does not change closure.