A. Gavrilov has several nice works studying the double exponential map and its properties ([1] and references therein).

Given a complete Riemannian manifold $(M,g)$ and a point $p \in M$, the double exponential map is defined as $\text{Exp}_p : T_p M \times T_p M \rightarrow M$ such that, if $v$ is of small norm, \begin{align} \text{Exp}_p (v,u) := \text{Exp}_{\text{Exp}_p (v)} (u_{\text{Exp}_p (v)}), \end{align} where $ u_{\text{Exp}_p (v)} $ is the vector of parallel transporting $u \in T_p M$ to $T_{\text{Exp}_p (v)} M$ along the minimizing geodesic connecting $p$ and $\text{Exp}_p (v)$.

Given $p \in T_p M$, one can define a smooth map $h_p : T_p M \times T_p M \rightarrow T_p M$ such that, for $v,u \in T_p M$ of small norm, it holds that \begin{align} \text{Exp}_p (h_p (v,u)) = \text{Exp}_p (v,u). \end{align}

A. Gavrilov derived a Taylor series approximation for $h_p (v,u)$, which is \begin{align} h_p (v,u) =& v + u + \frac{1}{6} R (u,v) v + \frac{1}{3} R (u,v) u + \frac{1}{12} \nabla_v R (u,v) v \\ &+ \frac{1}{24} \nabla_u R (u,v) v + \frac{5}{24} \nabla_v R (u,v) u + \frac{1}{12} \nabla_u R (u,v) u + \cdots, \tag{1} \end{align} where $R$ is the Riemann curvature tensor. This formula is Eq.3 in [1].

I strongly suspect that the above result can be thought as a corollary of the celebrated Baker-Campbell-Hausdorff-Dynkin (BCHD) formula. To link these two results, one can perhaps construct "coordinate vector fields" so that the BCHD formula can be applied. For $p$, within a small neighborhood $p$ (written $U_p$), let $X$ be the vector field such that $X_q = v_q$ for all $q \in U_p$, and let $Y$ be the vector field such that $Y_q = u_q$ for all $q \in U_p$. For vector fields $X$ and $Y$ over $U_p$, we have, if $\exp(Z) = \exp(X) \exp(Y)$, then it holds that \begin{align} Z = X + Y + \frac{1}{2} [X,Y] + \frac{1}{12} [ X ,[X,Y]] + \cdots. \tag{2} \end{align} For this pair of vector fields, $[X,Y] = 0$ and it seems (1) and (2) don't coincide.

I probably missed something very basic here, or I'm not using the right vector fields, or perhaps the results of Gavrilov cannot be thought of as a corollary of the BCHD formula. Any pointers?

References: [1] A. Gavrilov (2007), The Double Exponential Map and Covariant Derivation, Siberian Mathematical Journal.

A convenient link to [1]: https://www.academia.edu/7675502/The_double_exponential_map_and_covariant_derivation