Q: Does parallelizability by a frame of commuting vector fields imply those fields are coordinate vector fields of a global chart? If $M$ is a parallelizable manifold, is the following true?

If $(X_i)$ is a (global) frame of $M$ and $[X_i,X_j]=0$ for all $i,j$, then there is a global chart for which $X_i=\partial_i$.


A: No. The circle $\Bbb S^1 \subset \Bbb R^2$ is parallelizable by the frame containing the standard vector field, $X_{(x, y)} := -y \partial_x + x \partial_y$, but (because it is compact) it admits no global chart at all.
The vector field $X$ is sometimes abusively denoted $\partial_{\theta}$, but this question demonstrates how this notation can be misleading.
