Coordinate Jacobi condition for a Poisson bivector (${\omega }^{im}{\partial }_m{\omega }^{jk}+{\omega }^{jm}{\partial }_m{\omega }^{ki}+{\omega }^{km}{\partial }_m{\omega }^{ij}=0$)

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.