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Lie groups carrying a left invariant symplectic form (symplectic groups) are described in terms of semi-direct product of Lie groups or symplectic reduction and principal fiber bundles with affine fiber. We give a generalization of Medina and Revoy's symplectic double extension, which realizes a symplectic group as the reduction of another symplectic group. We show that every group obtained by this process carries an invariant Lagrangian foliation such that the affine structure defined by the simplectic form over each leaf is complete.