= Coordinate Jacobi condition for a Poisson bivector
{title2=$\omega^{im}\partial_m\omega^{jk}+\omega^{jm}\partial_m\omega^{ki}+\omega^{km}\partial_m\omega^{ij}=0$}
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.
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