I'm confused as to the relationship between Lie Groups and Lie algebras. From what I know, the Lie algebra of a Lie group is the space of tangent vectors which pass through the identity, together with Lie bracket defined by $[X,Y]=ad(X)(Y)$.
My question is, what actually is $ad(X)(Y)$ and what does it equal? I have found the following link: https://www.maths.gla.ac.uk/~gbellamy/lie.pdf and at the top of page 18, the author shows how $[X,X]=ad(X)(X)=0$, but I don't see how you can't use an analogous method to show $[X,Y]=ad(X)(Y)=0$.
Also, I have seen a lot on left-invariant vector fields, but what is the purpose if the Lie algebra of a Lie group is just the tangent space definition? Is this just an equivalent method of find a Lie algebra from a Lie group?