  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 $\mathbb{R}^n$. In the last section, for each flat left
invariant affine symplectic connection on the group of affine transformations
of the real line, describe for 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.
