Relations among Lie-series transformations and isomorphisms between free Lie algebras

We study the subgroup generated by the exponentials of formal Lie series. We show three different ways to represent elements of this subgroup. These elements induce Lie-series transformations. Relations among these family of transformations furnish algorithms of composition. Starting from the Lazard elimination theorem and the Witt's formula, we show isomorphisms between some submodules of free Lie algebras. Combining different results, we also show that the homogeneous terms of the Hausdorff series H(a,b) freely generate the free Lie algebra L(a,b) without a line.