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I don't understand several things.

First of all, why define "the Lie Algebra of G" this way? Obviously since G is a manifold, then the tangent spaces at every point in G ARE Lie Algebras; every tangent space is a vector space, and the vector space obviously is endowed with the Lie Bracket operation which takes vectors to vectors and obviously is anti-commutative and obeys the Jacobi Identities. Why go through this convoluted way of finding a Lie Algebra? Why not just pick a tangent space (say the tangent space at the identity element e) and call that "the Lie Algebra of G"?

Secondly, his statement "The left-invariant vector fields form a Lie Algebra...this is called the Lie Algebra of G", has me puzzled on what a Lie Algebra really is. I thought a Lie Algebra is any vector space which is endowed with a multiplication rule (the Lie bracket) which satisfies the 2 requirements of anti-commutivity and Jacobi-identity. A vector field does not live in any 1 vector space, it is a rule which defines at each point on the manifold a vector which lives in the tangent space to that point on the manifold. Therefore, wouldn't Shutz's statement that the vector FIELDS form a Lie Algebra, be more accurate if he said that they formed an infinity of Lie Algebras, with each Lie Algebra residing at each point on the manifold? I'm picturing here that every point P on the manifold G has it's own Lie Algebra.

Thirdly, Schutz mentions that every vector at e induces a unique left-invariant vector field. This statement is fine; however, what does this statement have to do with anything? Why is e special in the sense of defining Lie Algebras?

I've spent a long time trying to understand this to no avail...someone help please!