Consider a Lie group $G$ acting on itself. Every vector $v\in T_e G$ gives rise to a vector field $V$ associated to $v$:
$$ V_g=d(L_g)_e(v) $$which satisfies
$$ d(L_g)_h(V_h)=V_{gh}. \tag{1} $$All the vector fields satisfying (1) this are called left invariant vector fields, and the set of all of them is the Lie algebra $\mathfrak{g}$ of $G$. Observe that there is a one-to-one relation
$$ T_eG \to \mathfrak{g} $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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