Noncommutative differential calculus for Moyal subalgebras

We build a differential calculus for subalgebras of the Moyal algebra on R4 starting from a redundant differential calculus on the Moyal algebra, which is suitable for reduction. In some cases we find a frame of one-forms which allows to realize the complex of forms as a tensor product of the noncommutative subalgebras with the external algebra Λ∗Λ∗.