Milnor defines the rank of a smooth manifold $M$ as the maximum cardinality of a linearly independent set of vector fields on $M$ whose elements are pair wise commuting. In other words, the rank of $M$ is the greatest $k$ such that $\mathbb{R}^k$ acts locally freely on $M$.
It is known since the 60's that the rank of $S^3$ is $1$, a famous result due to Lima. I wonder if there has been similar results regarding other spheres since then. From what I could find, the problem of how many linearly independent vector fields a sphere supports (without any conditions on them being commutative) has been solved by Adams, and the number of l.i. fields is completely determined by the sphere's dimension and a suitable Radon-Hurwitz number. For a lot of spheres this number turns out to be $1$, so in particular the sphere's rank is $1$ as well.
To be more specific in my question, I'm particularly interested in results concerning the rank of $S^7$. Has it been calculated yet, or at least any bounds? Does the fact that the $7$-sphere is parallelisable imply anything about its rank?