  Binary operations on algebras of observables are studied in the quantum as
well as in the classical case. It is shown that certain natural compatibility
conditions with the associative product imply the properties which usually are
additionally required. In particular, it is proved that locality of a Loday
bracket on sections of a one-dimensional vector bundle forces skew-symmetry,
i.e. a local Lie algebra structure in the sense of A.A.Kirillov.

Differential Geometry;Operator Algebras