Let G be a Lie group. An analytic local multiplication μ at the identity element 1 of G is associated with a Lie algebra on the tangent space of G at 1. It is shown that μ is a geodesic homogeneous local left loop which is in projective relation with the group multiplication μ＾0 (Theorems 2.1 and 2.16), and that any geodesic homogeneous local left loop in projective relation with μ＾0 is given by such a μ (Theorem 3.3).