Given an exponential parameterization of a 3D rigid rotation $R\in SO(3)$ by the vector $v = (v_x, v_y, v_z)^T$ I would like to find its second derivatives at the point $v=(0,0,0)$.
Using the exponential parameterization rotations can be expressed using the matrix exponential $\exp$ and a function $J$ that maps vectors $v$ to skew-symmetric matrices corresponding to infinitesimal rotations such that $J(v) = [v]_{\times}$ are cross product matrices.
\begin{equation} R = \exp\{J(v)\} \end{equation}
In a paper [1] formulas for the first and second derivatives are provided. Results for the first derivatives are in alignment with well known results related to generators (see this question): \begin{equation} \left.\frac{\partial}{\partial v_x}R\,\right|_{v=0} = J(\hat x) \end{equation}
where $\hat{x}$ is the first unit vector.
Additionally the authors give a formula for the second derivatives without much explanation:
\begin{equation} \left.\frac{\partial}{\partial v_x \partial v_y}R\,\right|_{v=0} = \frac{1}{2}(J(\hat x)J(\hat y) + J(\hat y)J(\hat x)) \end{equation}
In [2] a formula for the second directional derivative is provided, when applied to directions corresponding to the unit vectors this gives the same result ($R_{xx}|_{v=0}=J(\hat{x})^2$).
My question is how the formula for the mixed derivatives could be rigorously derived, so that its correctness can be proven?
[1] C. J. Taylor and D. J. Kriegman, “Minimization on the Lie group SO (3) and related manifolds,” Yale University, 1994.
[2] M. Sarkis and K. Diepold, “Camera-pose estimation via projective Newton optimization on the manifold,” IEEE Trans. Image Process., vol. 21, no. 4, pp. 1729–1741, Apr. 2012.