Exponentials and one-parameter subgroups

Let $G$ be a Lie group with Lie algebra $\mathfrak g=T_eG$ .

Lemma (Exponential–one-parameter subgroup). For each $X\in\mathfrak g$ , there is a unique smooth one-parameter subgroup

\gamma_X:\mathbb R\to G

with $\gamma_X'(0)=X$ . Conversely, every one-parameter subgroup $\gamma$ is uniquely determined by $X=\gamma'(0)$ . Moreover, after the exponential map is defined by this correspondence,

\gamma_X(t)=\exp(tX).

Remarks

Context. This lemma packages the correspondence between elements of $\mathfrak g$ and flows of left-invariant vector fields: the curve $\gamma_X$ is the integral curve through $e$ of the left-invariant field determined by $X$ (compare one-parameter subgroups as integral curves). Locally, it is compatible with the fact that $\exp$ is a local diffeomorphism near 0.