[3rd in this expository series] By a change of coordinates, the unit ball is mapped onto the Siegel domain, where it is easy to spot the (Heisenberg) Lie group structure of the horocycles and the Riemannian metric obtained by restricting the Bergman-Kobayashi metric to them. Assuming the viewpoint of the visual metric (just a rescaling), the boundary of the Siegel domain inherits a (Heisenberg) sub-Riemannian structure.