[This question was asked on [MSE](https://math.stackexchange.com/questions/1788107/on-a-parallelizable-manifold-is-there-always-a-frame-satisfying-x-i-x-j-0), but got no answers, I thought it could be more appropriate here]


Let $M$ be a parallelizable manifold.


1. Is there always a global frame $(X\_i)$ such that $[X\_i,X\_j]=0$ for all $i,j$ ?
2. If the answer is no, what kind of obstruction there is to find such a frame ? and what kind of general (topological ?) condition on $M$ makes it possible to find such a frame ?



On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?
Suppose that $M$ is compact, such a frame exists implies that the commutative group $R^n$ acts transitively on it, this implies that $M$ is a torus. But a compact Lie group is parallelizable, so that is not always possible.


