A general form of Hamiltonian for vibronic Jahn-Teller systems is derived on the basis of adiabatic approximation and group theory. The electronic operators and active Jahn-Teller modes appearing in a vibronic system are also discussed. Further calculations of excited states in minima are carried out using unitary transformation method and energy minimization procedure. The results of energy splitting for an electronic triplet Jahn-Teller system are analyzed and compared with particular reference to tetrahedral and its related crystal systems. It is shown that the lift of electronic degeneracy can be quantitatively described by the decomposition of irreducible representations of related group and subgroups.