Can Lie Algebra Structures Be Extended to Tangent Spaces Near the Identity?

[Pietjuh]

I've been studying some things about Lie algebras and I've got some questions about it.

We know that for a given Lie group G, the tangent space at the identity has the structure of a Lie algebra, where the Lie bracket is given by [X,Y]=ad(X)(Y). This map ad is the differential at e of the map Ad which is defined by Ad(g)(X) = gXg^{-1}. Now I'm wondering whether it's possible to associate a Lie algebra structure on tangent spaces at points near the identity element, for example in an open set containing e, or for points in the same connected component in which e lies. I don't think you could use the map Ad for it, but perhaps some slight modification is possible??

Thanks in advance! :)

 

[Cexy]

I don't see an immediate problem - rather than taking the definition of the Lie algebra L(G) associated with the group G to be the tangent space at the identity, just take it to be the algebra of vector fields on G. Then any vector field $$V_X(g)$$ which is generated by applying the push-forward $$L_{g*}$$ to the vector $$X \in T_eG$$ can equally well be generated by first applying $$L_{p^{-1}*}$$ to a vector $$Y \in T_pG$$ and then applying $$L_{g*}$$.