We characterize the symmetry algebra of the generic superintegrable system
on a pseudo-sphere corresponding to the homogeneous space SO(p, q + 1)
/SO(p, q) where p+ q = N,N ∈N. These symmetries occur both in quantum
as well as in classical systems in various contexts, so they are quite important in
physics.We show that this algebra is independent of the signature (p, q + 1) of
the metric and that it is the same as the Racah algebraR(N + 1). The spectrum
obtained from R(N + 1) via the Daskaloyannis method depends on undetermined
signs that can be associated to the signatures. Two examples are worked
out explicitly for the cases SO(2, 1)/SO(2) and SO(3)/SO(2) where it is shown
that their spectrum obtained by means of separation of variables coincide with
particular choices of the signs, corresponding to the specific signatures, of the
spectrum for the symmetry algebra R(3).
