The rank of an smooth manifold M is defined by Milnor, as follows:

"The maximum number of independent commuting vector fields on M"

For example it is well known that the rank of $S^{3}$ is 1 (Lima, On commuting vector fields on S^3, Annals of math,1965)

One can consider a generalization as follows:

The Frobenius rank of an n- manifold is:

The maximum number $k< n$ such that there are k independent smooth vector fields $X_{1},X_{2},\ldots,X_{k}$, globally defined on $M$, such that $[X_{i}, X_{j}](p) \in \text{Span}(X_{1}(p),\ldots X_{k}(p))$, for all $p\in M$ and for all $i,j\in \{1,2,\ldots,k\}$.

Now the question:

What is the Frobenius rank of $S^{3}$ or $S^{7}$?

In fact this question searchs for the maximum number $k$ such that $M$ admits a $k$ dimensional foliation which distribution can be generated by k global independent vector fields?

The motivation for this question is the question "Can the Reeb foliation be generated by two independent global vector fields?"