Poisson bivector ($\omega =\frac{1}{2}{\omega }^{ij}{\partial }_i\wedge {\partial }_j$)

A smooth antisymmetric contravariant two-tensor defines a biderivation $\{F,G\}=\omega (dF,dG)$. It is a Poisson bivector when this bracket satisfies the Jacobi identity. Its rank may vary; where it is nondegenerate it inverts to a symplectic form. The coordinate Jacobi condition for a Poisson bivector makes the integrability constraint explicit.

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.

The Jacobi identity for a Poisson bivector reduces to the displayed cyclic identity in every local chart. Necessity follows by applying the bracket to coordinate functions. Sufficiency follows because the second-derivative terms in an arbitrary Jacobiator cancel by antisymmetry, leaving this coefficient multiplying first derivatives of the three functions.