The following problem has recently come up in the course of a project I'm working on, and I would be very appreciative if someone could point me in the right direction:

Let $X \colon I \to G$ be a smooth curve taking values in a Lie group $G$ with a surjective exponential map, and define another curve $v\colon I \to T\_e(G)$ according to $v(t) \triangleq \omega\_{X(t)}\left( \dot{x}(t) \right)$, where $\omega$ is the Maurer-Cartan form on $G$. Then intuitively, $v(t)$ represents a kind of "local" measurement of the velocity $\dot{X}(t)$, expressed in a local, "co-moving" coordinate system that is rigidly attached to the point $X(t)$ itself.

Given access to $v(t)$ and an initial condition of the form $X(t\_0) = X\_{t\_0} \in G$ for some $t\_0 \in I$, I would like to determine a curve $\gamma \colon I \to T\_e(G)$ such that $X = \exp \circ \gamma$.