This paper deals with the low-dimensional dynamics in the general non-Abelian Kuramoto model of mutually interacting generalized oscillators on the 3-sphere. If all oscillators have identical intrinsic generalized frequencies and the coupling is global, the dynamics is fully determined by several global variables. We state that these generalized oscillators evolve by the action of the group of (quaternionic) Möbius transformations that preserve . The global variables satisfy a certain system of quaternion-valued ordinary differential equations, that is an extension of the Watanabe-Strogatz system. If the initial distribution of oscillators is uniform on , additional symmetries arise and the dynamics can be restricted further to invariant submanifolds of (real) dimension four.

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