Summary: The reductions of the free geodesic motion on a noncompact 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 \(\text{SU} (n + 1, n)\) and from \(\text{SU} (n, n)\). This generalization of the Olshanetsky-Perelomov derivation of the \(BC_{n}\) model with two independent coupling constants from the geodesics on \(G / G_{+}\) with \(G = \text{SU} (n + 1, n)\) 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 noncompact as well as for compact \(G\).