Assuming special relativity and Hamiltonian particle dynamics for a noncanonical Poisson bracket, the Jacobi identity is shown to have nontrivial physical consequences, including the homogeneous Maxwell equations and the geodesic law of motion of curved spacetime.
The right-hand side of the Jacobi identity: to be naught or not to be?
The Jacobi identity for graded-commutative variational Schouten bracket revisited
Jacobi Identity for Vertex Algebras in Higher Dimensions
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