Electromagnetic-field TE/TM decomposition theory, well known for fields in isotropic and uniaxially anisotropic media and their extension to uniaxial bi-anisotropic media is generalized to cover all bi-anisotropic media. General conditions for the medium parameter dyadics are derived for the decomposition to be possible. Some decomposable special bi-anisotropic media are considered as examples of applying the general conditions. Previously known cases are seen to fall under the conditions derived. The decomposition theory can be applied to splitting an electromagnetic problem in two simpler problems.