I've seen that from a tangent vector of the tangent space at identity element of a Lie group can be made into a left-invariant vector field: $ X_g = l_{g*} (X_e) $ in which, $X_e$ is a tangent vector at identity and $X_g$ is a tangent vector at some $g \in G$, the Lie group. Many sources then say it's enough to consider the tangent space at identity.

An exponential map will map from Lie algebra (a tangent vector in a tangent vector space at the identity) to a Lie group $G$. My question is: why is it enough to only consider the tangent space at identity and what happens if one decides to use the tangent space at some element rather than the identity?

I've also seen that the exponential map restricts to a diffeomorphism from some neighborhood of 0 in ${\displaystyle {\mathfrak {g}}}$ to a neighborhood of 1 in $G$ (link). In this case, wouldn't there exists several members of $G$ "far away" from neighborhood of 1 that makes the exponential map invalid? And, can all member of $G$ be mapped onto by the exponential map from some tangent vector $X_e$ at identity?

I'm new to this field thus I believe I've messed up the definitions so bad. Thank you for your help.



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