  We consider trivializations of second iterated bundles of a Lie group that
preserve lifted group structures. With such a trivialization, we elaborate
Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and,
both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which
is tangent of cotangent bundle of Lie group. We present all possible Poisson,
symplectic and Lagrangian reductions of spaces and corresponding dynamics on
them. In particular, reduction of Lagrangian dynamics on second iterated
tangent bundle includes reduction of dynamics on second order tangent bundle.
