We know that Lie Groups are parallelizable, I was looking for a version of the converse and came across this: https://books.google.com/books?id=w4bhBwAAQBAJ&pg=PA115 in Introduction to Smooth Manifolds by John M. Lee (1st edition, Springer, 2003).

As says in the link : a remarkable theorem by Joseph Wolf says that the only compact,simply connected manifolds that are parallelizable are products of Lie groups and copies of $S^7$

But, I am not being able to find this proof, If anyone can give a proof or a reference that would be much appreciated. Thanks in advance.

$\textbf{Update} $: As discussed in the comments the following result in Lee's book isn't exactly correct. Anyway these are the Wolf's papers mentioned in the link:

[Wol71] Joseph A. Wolf. On the geometry and classification of absolute parallelisms I. J. Differential Geometry, 6:317-342, 1971. projecteuclid, author's website;

[Wol72] Joseph A. Wolf. On the geometry and classification of absolute parallelisms II. J. Differential Geometry, 7:19-44, 1972. projecteuclid, author's website

The original question about conditions which put a multiplicative or a group structure on a parallelizable manifold still remains unanswered. So any help would be appreciated.