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From classical and quantum mechanics we abstract the concept of a two-product algebra. One of its products is left unspecified; the other is a Lie product and a derivation with respect to the first. From composition of physical systems we abstract the concept of composition classes of such two-product algebras, each class being a semigroup with a unit. We show that the requirement of mutual consistency of the algebraic and the semigroup structures completely determines both the composition classes and the twoproduct algebras they consist of. The solutions are labelled by a single parameter which in the physical case is proportional to the square of the quantum of action.

In classical and in quantum mechanics physical variables play a dual role as observables and as generators of infinitesimal transformations in the invariance groups. We show that if the Lie algebra of generators is central simple, the observable-generator duality restricts the structure of the algebra of observables to two cases: a commutative, associative algebra as in classical mechanics, or a central simple special Jordan algebra as in quantum mechanics.