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](https://doi.org/10.1007/s005260050142).}},
           year = {1999},
           journal = {{Calculus of Variations}},
           volume = {9},
           pages = {277--296}.

I would like to include a one-page LaTeX article and a screen shot to help explain my question but I don't know how to do that.