Conformal flatness, non-Abelian Kaluza–Klein reduction and quaternions

The non-Abelian Kaluza–Klein reduction of conformally flat spaces is considered for arbitrary dimensions and signatures. The corresponding equations are particularly elegant when the internal space supports a global Killing parallelization. Assuming this imposes the generalized ‘spacetime’ to be maximally symmetric with holonomy in the unitary quaternionic group Sp(d/4)Sp(d/4). Recalling an analogous result for the complex case, we conclude that all special manifolds with constant properly ‘holonomy-related’ sectional curvature, are in natural correspondence with conformally flat, possibly non-Abelian, Kaluza–Klein spaces.