Algebraic Properties of Covariant Derivative and Composition of Exponential Maps

We consider the problem of calculating the Taylor series for a function $h_x\colon T_xX\times T_xX\to T_xX$ defined by the composition of exponential maps, where $X$ is a smooth manifold with affine connection and $x\in X$. We show that the homogeneous summands of such a series can be derived by applying the Lie bracket and covariant derivative to the arguments of the function which are extended to vector fields.