I'm working with some SU(n) groups, primarily U(1), SU(2) and SU(3) at the moment. These have well-known Lie algebras which allow one to express representations of the groups in terms of the corresponding reps of the Lie algebra, as the matrix exponential of sums over its generators with n real coefficients. Since these groups are also n-dimensional smooth manifolds those n coefficients might also be thought of as a set of n real variables for the the points of the group's manifold, albeit presumably for only some portion of the manifold in the case of any single representation since it represents only a subset of the group.
Topologically these group manifolds have long been known to be various fiber bundles of spheres. For the three noted above these are S1, S3, and the unique non-trivial bundle S5 X S3 respectively. Explicit expressions for the matrices of SU(3) in terms of the points of its non-trivial fiber bundle have been obtained by Aguilar & Socolovsky in https://arxiv.org/abs/hep-th/9812006.
My question is what relationship(s), if any, have been established between such topological and Lie algebra expressions for SU(n) matrices?