Distinction between Lie Algebra of differentiable manifolds and Lie Algebra of Lie Group

Question

Is it true that we can always define a Lie Algebra for any differentiable manifold by using the bracket $[X,Y] = XY - YX$ on all vector fields defined on the manifold, but in order to define a Lie Algebra associated with a Lie Group, we must restrict this bracket to just left-invariant vector fields on a (Lie Group's) manifold (which is believable since the left-invariant definition makes use of the group action)?

Answer 1

The Lie algebra of all vector fields defined on a manifold is infinite-dimensional. In the case of a $n$-dimensional Lie group $G$, we define its Lie algebra as the Lie algebra $\mathfrak g$ of all left-invariant vector fields because then we then get a $n$-dimensional Lie algebra. And because then the Lie algebra structure of $\mathfrak g$ has something to do with the group structure on $G$.