I`ve already asked this in the comments below this article ...
Does Kaluza-Klein theory successfully unify GR and EM? Why can't it be extended to the Standard Model gauge group?
As a quick disclaimer, I thought this might be a better place to ask than Physics.SE. I already searched there with "kaluza" and "klein" keywords to find an answer, but without luck. As background, ...
I am thinking about Kaluza Klein theory in the 3 dimensional lens spaces. These have an isometry group SU(2)xU(1), generically, and in some way interpolate between the extreme cases of manifolds $S^2 ...