Electromagnetic sources in a uniaxial anisotropic medium are known to be decomposable in two parts producing fields with TE and TM polarization with respect to the axis of the medium. Also, an expression for the Green dyadic in the same medium has been derived. In this study, these two solution principles are brought together by finding the TE/TM decomposition of the Green dyadic. This is done by finding the solution for the ensuing fourth-order partial differential equations in analytic form. In addition to being conceptually attractive, the new decomposed expression is helpful in solving electromagnetic problems with either TE or TM character, in which case only one of two components of the Green dyadic is needed.