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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.

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    The set of all vector fields on a Lie group (or, more generally, a Riemannian manifold) is an infinite dimensional vector space. If one consider the set of left-invariant vector fields the one gets a finite dimensional vector space. Moreover, it is isomorphic to the tangent space at $e\in G.$ – mfl Mar 17 '15 at 23:22
  • Also, the set of all vector fields is certainly quite interesting but its definition does not use in any way the multiplication on $G$. (It is not the Lie algebra of $G$ but the Lie algebra, in a suitable sense, of the diffeomorphism group $\text{Diff}(G)$.) – Qiaochu Yuan Mar 18 '15 at 04:51

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