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A vector field X is said to be left invariant if it commutes with left translation: Define L_g[f](x)= f(gx), then a vector field is left invariant if X L_g = L_g X for all g in G.
The set of all vector fields on an analytic manifold is a Lie algebra. The subalgebra of all left invariant vector fields is called the Lie algebra associated to G, and is usually denoted by a gothic g.