A symplectic Lie group is a Lie group $G$ together with a left invariant symplectic 2-form . Every symplectic Lie group admits a natural affine structure determined by its symplectic form. The symplectic double extension gives a procedure for constructing new symplectic Lie groups from old ones. In this talk, we will focus on certain symplectic Lie groups having an exact symplectic 2-form that can be derived by symplectic reduction or double extension. We study the geometry of these symplectic manifolds. In particular we will show that these Lie groups carry two transversal invariant Lagrangian foliations with affine and closed leaves.