Stability of equilibria for the SO(4) free rigid body

It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras which on a regular adjoint orbit give three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in so(3). The nonlinear stability and instability of these equilibria is determined. In addition to these equilibria there are others that come in curves. It is shown that these curves of equilibria are nonlinearly stable in the sense that the only possible drift in is the direction of the curve itself.