I'm looking for a book where the Erlangen program is carried out on some example groups with explicit computations.

What I mean by "carrying out Erlangen program" is picking a specific group (say SO(n+1)) and some subgroup (say SO(n)), finding *all* invariants and their relations (and the relations between these relations and so on), and see what it means for the geometry of the natural space our group acts on (say n-sphere).
So all the work should be done in the group (including finding good parametrizations for geodesics and this kind of things) rather than in the usual space.

Anyone knows of a good reference?