  We provide an alternative method for obtaining of compatible Poisson
structures on Lie groups by means of the adjoint representations of Lie
algebras. In this way, we calculate some compatible Poisson structures on four
dimensional and nilpotent six dimensional symplectic real Lie groups. Then
using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems with four
dimensional and nilpotent six dimensional symplectic real Lie groups as phase
spaces.
