Define the trajectory $\gamma^f$ to be the solution to the dynamics $\dot{x} = f(x)$ on an open interval containing $0$ and assume regularity conditions on the manifold that $x$ lies in and the vector field $f$ so that it exists and is unique. The trajectory along $[f, g]$ where $[f, g]$ is the Lie bracket between two vector fields $f, g$ is given by the trajectory expressed as compositions \begin{equation} \gamma^{-g} \circ \gamma^{-f} \circ\gamma^g \circ \gamma^f . \end{equation} Similarly, we can obtain an expression for the trajectory along $[f, [f, g]]$ which contains the composition of 10 terms of only $\gamma^f, \gamma^g$ and their inverses, and the number of compositions increases rapidly with the number of nested Lie brackets. Is there a way to simplify the expression so that the trajectory can be expressed as a composition of fewer elements, i.e. can we express $\gamma^{[f, [f, g]^n]}$ as fewer than $O(2^n)$ terms involving $\gamma^f, \gamma^g$ or their inverses without using any of $\gamma^{[f, [f, g]]}, \dots, \gamma^{[f, [f, g]^{n-1}]}$ which trivially simplifies calculations, where $[f, g]^n$ is the nested Lie bracket $[f, [f, g]], \dots$?
I'm expecting that perhaps the Baker–Campbell–Hausdorff formula, elementary properties of the Lie bracket, and/or mild constraints on the manifold/ODE allow for an expression that involves much fewer compositions.