I'm trying to read section 3 in J. Jost and Y.L. Xin [JX], This section has to do with the geometry of Grassmannians. I have a question that comes out of the derivation at the top of page 283 in that paper. Let $M$ be a locally symmetric Riemannian manifold. Let $c$ be a geodesic in $M$ and let $J$ be a Jacobi field along $c$. Let $\cdot$ be the derivative (differential) of $c$ and assume $J$ is orthogonal to $\cdot$. Under these hypotheses is it the case that the Lie bracket, $[\cdot, J]$, is $0$? If so, why?

[JX] title = {{Bernstein type theorems for higher codimension.}}, year = {1999}, journal = {{Calculus of Variations}}, volume = {9}, pages = {277--296}.