Parallelizable spheres are H-spaces

Adams's paper On the Non-Existence of Elements of Hopf Invariant One famously includes the following diagram of implications in the introduction:

Implications in the Hopf Invariant One problem

I want to focus on the implication (5): $S^{n-1}$ being parallelizable, perhaps with some extraordinary differentiable structure, implies that it is an H-space. Adams attributes this to "A. Dold, in answer to a question of A. Borel," but gives no further reference.

It is not terribly difficult to show this implication: it is carried out, for example, in Haynes Miller's notes on Vector Fields on Spheres, Lecture 1. My question is instead one of history:

Where does this implication first appear in the literature? Is Adams correct about it being due to Dold?

Any pointers to where I can best start sleuthing would be appreciated.

• A proof appears as Lemma 1 of "A Note on Tangent Bundles" by Kenichi Shiraiwa, Nagoya Mathematical Journal 1967 doi.org/10.1017/S0027763000024363 (pdf available). He also comments that he could not find a reference. – j.c. Jul 18 '17 at 16:56