Since the set of all vector fields $V$ on a Lie group $G$ forms a vector space, one can impose algebraic structure (a Lie algebra) by defining the bracket $[\cdot,\cdot]$ between these vector fields.
If it's possible to associate a Lie algebra in the manner described above, why is necessary to consider a subspace $V$, i.e., the set of all left-invariant vector fields, to build up the Lie algebra? If there's something wrong in my reasoning in the last paragraph, please indicate it.