  The reductions of the free geodesic motion on a non-compact simple Lie group
G based on the $G_+ \times G_+$ symmetry given by left- and right
multiplications for a maximal compact subgroup $G_+ \subset G$ are
investigated. At generic values of the momentum map this leads to (new) spin
Calogero type models. At some special values the `spin' degrees of freedom are
absent and we obtain the standard $BC_n$ Sutherland model with three
independent coupling constants from SU(n+1,n) and from SU(n,n). This
generalization of the Olshanetsky-Perelomov derivation of the $BC_n$ model with
two independent coupling constants from the geodesics on $G/G_+$ with
G=SU(n+1,n) relies on fixing the right-handed momentum to a non-zero character
of $G_+$. The reductions considered permit further generalizations and work at
the quantized level, too, for non-compact as well as for compact G.
