Internal Symmetry and Lorentz Invariance

The notion of a rigorous internal symmetry implies an over-all symmetry group $G$ that contains the inhomogeneous Lorentz group as a proper subgroup. Such a rigorous symmetry does not automatically require degenerate mass multiplets. But over-all symmetry groups that are compatible with mass splittings are severely restricted as follows. Assume the generators of $G$ are the Lorentz generators and the generators of either a semisimple or a compact Lie group. If the Cartan subalgebra of its semisimple part is Lorentz invariant, then all the generators of the internal symmetry are Lorentz invariant and therefore there can be no mass splitting. In particular, if the internal symmetry is SU(3) and $T_z$ and $Y$ are Lorentz invariant, then all the generators of SU(3) are Lorentz invariant.