The topological completion of a bilinear form

Let M=Mn,m be the Euclidean space Rp equipped with a symmetric bilinear form BM of rank p=n+m and signature n−m. We compactify M so that Mc is homogeneous and has as its group of isometries a Lie group whose dimension is the dimension of M plus 2p+1. We observe that Mc is in two ways the total space of a non-trivial sphere bundle with base space real projective space. The compactification is well understood in the classical case when M is Minkowski space. The contribution here is to observe that the construction works generally and that it admits a natural bundle description.