Plane-wave propagation is studied in the most general bianisotropic medium in which the electromagnetic wave can be split into two eigenwaves with certain polarizations depending on two arbitrary directions in space. Hence the whole problem splits into two parts. The dispersion dyadics of the materials seen by the eigenwaves are derived and the refractive indices are solved for arbitrary direction of propagation. Also, a general expression for the eigenpolarizations is found. The theory is applied to some cases fitting the requirements of the media.