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We theoretically examine the energy spectrum and anisotropic g-factor of an elliptic quantum dot in the presence of strong spin-orbit (SO) interaction. The Kramers' degeneracy of energy levels in the quantum dot is lifted by applying a magnetic field. The splitting of the levels reflects the wavefunctions through the SO interaction and hence depends on the direction of the magnetic field. We find that the g-factor becomes largely anisotropic, in accordance with experimental results. The anisotropy significantly differs from level to level.