I know that the span of the vectors in the tangent space at the identity of a Lie Group is isomorphic to the span of the left-invariant vector fields over the Lie Group. It seems to me (a physicist) so much easier to consider a Lie Algebra by this $T_{e}$ tangent space at the identity, so I'm wondering why groups resources define Lie Algebras using left-invariant vector fields at all?
My only guess is that left-invariant vector fields are required in order to uniquely specify the integral curves used in defining the exponential map?
Best Answer
To expand out Nate's comment, if we just define a Lie algebra as the tangent space of a Lie group at the identity it isn't clear why it should have the structure of a Lie bracket on it and why other manifolds don't have this property.
So we note that vector fields on a manifold have a Lie bracket structure intrinsically ($XY$ is not a vector field but $[X,Y]=XY-YX$ is where we are treating vector fields as derivations and $XY$ is their composition). Then on a Lie group we can define left invariant vector fields and the Lie bracket on all vector fields gives a natural Lie bracket on this subspace. Then we identify the tangent space at the identity with the space of left invariant vector fields and this equips the tangent space with this Lie bracket.