For the subgroups K'AN of SO(p,q) acting linearly on R^{p,q}, every orbit outside a p-dimensional degenerate subspace W^p is independent of the choice of K', while orbits on W^p depend on K', and the nilpotent factor N alone never acts with codimension-one orbits.
Berard-Bergery, Sur de nouvells vari ´et´e riemanniennes d’Einstein,Inst
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On Cohomogeneity One Linear Actions On Pseudo-euclidean Space $\mathbb{R}^{p,q}$
For the subgroups K'AN of SO(p,q) acting linearly on R^{p,q}, every orbit outside a p-dimensional degenerate subspace W^p is independent of the choice of K', while orbits on W^p depend on K', and the nilpotent factor N alone never acts with codimension-one orbits.