Hamiltonian systems related to invariant Metrics

Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation which acts on the Lie algebra. By inducing the canonical symplectic structure of the coadjoint orbits to the orbits of the new representation, we study Hamiltonians on the orbits by determining the Hamiltonian vector fields for invariant functions and we find commuting conditions under the corresponding Lie-Poisson bracket.