Relatively free nilpotent torsion-free groups and their Lie algebras

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relatively free in some variety of nilpotent groups, is Magnus and certain Lie algebras associated to H are isomorphic. This isomorphism is extended to relatively free residually torsion-free nilpotent groups. Finally, we give an example that demonstrates that this is not always the case with finitely generated Magnus nilpotent groups.