One-parameter subgroups as integral curves

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

Fix $X\in\mathfrak g$, and let $X^L$ be the corresponding left-invariant vector field on $G$ (obtained by translating $X\in T_eG$ via left translations).

1. The integral curve of $X^L$ starting at the identity is the one-parameter subgroup
   $$t\longmapsto \exp(tX),$$

where $\exp$ is the exponential map. In particular, $\exp(tX)$ solves the ODE $g'(t)=(X^L)_{g(t)}$ with $g(0)=e$.

1. More generally, the integral curve of $X^L$ starting at $g_0\in G$ is
   $$t\longmapsto g_0\,\exp(tX).$$

There is an analogous statement for the right-invariant vector field $X^R$, whose integral curves are $t\mapsto \exp(tX)\,g_0$.