In Ginsparg's Applied Conformal Field Theory (http://arxiv.org/abs/hep-th/9108028, on the bottom of p. 5) the following remark is made:
Indeed the conformal group admits a nice realization acting on $\mathbb{R}^{p,q}$, stereographically projected to $S^{p,q}$ and embedded in the light-cone of $\mathbb{R}^{p+1,q+1}$.
What does this mean?
Your last $S$ should be an $R$.
The conformal group on is the set of transformations of $R^{p+q}$ that preserve angles ona $R^{p, q}$ is a Euclidean space with $p$ normal dimensions and $q$ imaginary ones; $S^{p,q}$ is presumably the unit sphere in this space; stereographic projections from the real space to the sphere will preserve angles. Then for each point on the sphere you can associate a future-pointing light ray passing through points in $R^{p+1,q+1}$, so each of the points on the sphere can be embedded in the larger light cone.
