In this paper we exhibit a family of flat left invariant affine structures on the double Lie group of the oscillator Lie group of dimension 4, associated to each solution of classical Yang-Baxter equation given by Boucetta and Medina.On the other hand, using Koszul's method, we prove the existence of an immersion of Lie groups between the group of affine transformations of a flat affine and simply connected manifold and the classical group of affine transformations of R n .In the last section, for each flat left invariant affine symplectic connection on the group of affine transformations of the real line, describe by Medina-Saldarriaga-Giraldo, we determine the affine symplectomorphisms.Finally we exhibit the Hess connection, associated to a Lagrangian bi-foliation, which is flat left invariant affine.