Commuting coordinate basis

Every pair of coordinate basis fields commutes:Therefore a frame containing two fields with nonzero Lie bracket of vector fields cannot be induced by a coordinate system.

On the punctured Euclidean plane, the orthonormal polar frame isIts bracket is $[\hat{r},\hat{\theta }]=-\hat{\theta }/r$, so it is not a coordinate basis even though $({\partial }_r,{\partial }_{\theta })$ is one.