I was reading these notes and in the fifth page it is said that:
Given a (smooth) manifold of dimension $n$, $M$, $M$ is parallelizable iff $M$ is the product of a Lie group and some number of copies of $\mathbb{S}^7$.
If $M$ is the product of a Lie group and some number of copies of $\mathbb{S}^7$, then $M$ is parallelizable since it is the product of parallelizable manifolds. However, I don't see the "only if" part of the statement. So any help would be appreciated, both a reference or an explicit argument or idea on how to prove that.
Just for the sake of completeness, I say that the manifold $M$ is parallelizale if $TM \cong M \times R^n$, or, equivalently, if it admits $n$ linearly independent vector fields.
Just for future reference, the statement is false. See, for example this question.