Does the existence of the cross product relate to the triviality of $TS^n$?

Question

The dimensions $n$ for which there exists a cross product in $\mathbb{R}^n$ are exactly those for which the tangent bundle $TS^n$ of the $n$-sphere is trivial (i.e. $TS^n = S^n \times \mathbb{R}^n$): $n= (0),1,3,7$. Is there an underlying reason for this? If so, are there any other situations in which these numbers arise?

Answer 1

The cross products on those particular $\Bbb R^n$ is essentially due to the fact that normed division algebras only exist in the corresponding $\Bbb R^{n+1}$ (Hurwitz's theorem). It seems that the triviality of tangent bundles on these particular $S^n$ is due to the Lie group structure on them, which I believe also follows from Hurwitz's theorem.