$M$ is a manifold, $X$ and $Y$ are tangent vectors in $T_xM$. Is it always possible to extend $X$ and $Y$ to local vector fields $U$ and $V$ around $x$ so that $[U,V]=0$ hold on some neighborhood of $x$? Or at least a weaker version: so that $[U,V]=0$ at $x$.
This arises from note 1 in the Wikipedia Exterior covariant derivative article, where some special tangent vectors are extended to special commuting vector fields using some specific construction.
Assume for simplicity that $M$ is 2-dimensional, and that $X$ and $Y$ are linearly independent. Choose coordinates $(u,v)$ in a neighborhood of the point $x$ in such a way that at the point, one has $X=\frac{\partial}{\partial u}$ and $Y=\frac{\partial}{\partial v}$. Since the coordinate vector fields $\frac{\partial}{\partial u}, \frac{\partial}{\partial v}$ always commute, we thereby obtain the necessary extension $U=\frac{\partial}{\partial u}$ and similarly for $V$. In higher dimension the argument is similar.
