Pair symmetry from the normal-coordinate curvature formula ($R_{ij,kl}=R_{kl,ij}$)

With convention $R(X,Y)={\nabla }_{[X,Y]}-[{\nabla }_X,{\nabla }_Y]$, and $R_{ij,kl}=g(R({\partial }_i,{\partial }_j){\partial }_k,{\partial }_l)$, normal coordinates giveat their center. Symmetry of the metric and commutation of second derivatives prove pair exchange. The opposite curvature convention changes the formula's overall sign but preserves all these algebraic symmetries.