 A popular (Fourier) integral representation for it, introduced by George Baker is

Each correspondence map from phase space to Hilbert space induces a characteristic "Moyal" bracket (such as the one illustrated here for the Weyl map).All such Moyal brackets are "formally equivalent" among themselves, in accordance with a systematic theory.The Moyal bracket specifies the eponymous infinite-dimensional 
Lie algebra—it is antisymmetric in its arguments and , and satisfies the Jacobi identity.