What is the advantage of defining Lie Algebras by left-invariant vector fields of a Lie Group?

Solution 1:

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). 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.