A reachability problem for the Heisenberg group

The reachability problem is a control system problem on a given manifold. Taking the three-dimensional Heisenberg group as our manifold with certain controls placed on a system, we want to know what sets of point on the manifold one can reach. In (jrder to do this, the basic form of a horizontal vector on a principal bundle over the manifold needs to be calculated. By generating a Lie algebra, using horizontal vectors on a principal bundle the reachable set of the system can be found.