[This question was asked on MSE, but got no answers, I thought it could be more appropriate here]

Let $M$ be a parallelizable manifold.

Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$ for all $i,j$ ?

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 ?

rankof the manifold. Generally this number is pretty small. $\endgroup$ – Ryan Budney May 17 '16 at 19:28