I am glad Glen mentioned *hyperbolic spaces*. What struck me as strange about choosing the special orthogonal groups was the positive curvature. That is, while we get a clear image of the non-commutativity of the Lie bracket, path-dependence sorta fails to do what I assume Jochen would want from it. I mean, two completely different paths can end up at the same point on the sphere, even if holonomy leaves the transported vector pointing in different directions. What I might have in mind, instead, is something more like the divergence of paths on a negatively curved geometry, rather than a difference in orientation at a shared endpoint.
But I have already raised what I think are larger objections earlier, and don't want to fight another's LLM party.
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