  In field theories one often works with the functionals which are integrals of
some densities. These densities are defined up to divergence terms (boundary
terms). A Poisson bracket of two functionals is also a functional, i.e., an
integral of a density. Suppose the divergence term in the density of the
Poisson bracket be fixed so that it becomes a bilinear form of densities of two
functionals. Then the left-hand side of the Jacobi identity written in terms of
densities is not necessarily zero but a divergence of a trilinear form. The
question is: what can be said about this trilinear form, what kind of a higher
Jacobi identity (involving four fields) it enjoys? Two examples whose origin is
the theory of integrable systems are given.
