The aim of this paper is to provide the reader with a rigorous, well-structured, and complete study of the most general decomposable bianisotropic media. These media were introduced elsewhere. We will give full expressions for the factorization of the Helmholtz determinant operator, for the equivalent media, for the Green dyadics of these equivalent media, and for the generalized Hertz potentials. This will be done efficiently using a six-vector formalism. Finally, we will treat, as an example, an anisotropic medium that has not been treated before.