The completely diagonalized Hamiltonian matrixes of 120 orders of 3d3/3d7 ions configuration in the trigonal symmetry were established by irreducible representation method and group theory, taking into account spin-orbit interaction, spin-spin interaction and spin-other-orbit interaction. The ground-state energy levels, the zero-field splitting parameter were calculated with the matrixes. The contributions of spin doublets for the ground-state energy levels were studied. The values of theoretical calculation were consistent with the experimental ones. The results showed that the contributions of doublets could not be neglected. On these bases, the effects of spin-spin interaction and spin-other-orbit interaction on the fine structures of spectra and zero-field splitting parameter of YAG∶Cr3+ crystal were further studied, and we found that the influence of spin-spin interaction and spin-other-orbit interaction on the fine structures of spectra and the zero-field splitting parameter could not be neglected. By comparing the values of theoretical calculation with the experiments, the existence of Jahn-Teller effect in the YAG:Cr3+ crystal spectra was confirmed.