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Obviously $Spin(4)$ is natural and in fact the eigenspinors of the Dirac operator are in one-to-one correspondence with this group, and from $Spin(4) = SU(2) \times SU(2)$ we have the naturality of $SU(2)$.  The most interesting group that occurs as a nontrivial gauge group is the $SU(3)$ for the strong force.  A bigger group $SU(4) = Spin(6)$ but $SU(3)$ is not identical to a spin group.  It is worthwhile exploring natural Lie groups to understand if the symmetry groups occurring in nature can be described without additional extraneous data such as a principal bundle.  This may seem like a purely aesthetic concern but it is valuable for the S4 picture where the simplest possibility is that particles are described by eigenspinors of the Dirac operator.