Constructing a parallelization of the 7-sphere.

Question

I would like to show that $S^7$, the 7-sphere, is a parallelizable manifold. Let $\mathcal{O}$ be the octonions, the normed division algebra (noncommutative, nonassociative) over $H\times H$, where $H$ is the quaternion algebra. Using $\mathcal{O}$ we can define a parallelization of $S^7$. Would someone explain this construction?

Answer 1

The sphere $S^7$ can be identified with the unit octonions (the octonions of norm 1). If $v$ is any nonzero tangent vector at the identity element $1 \in S^7$, then we can define a continuous nonzero vector field on $S^7$ by assigning to each point $x \in S^7$, the vector $v$ "multiplied" by $x$ using the octonion multiplication. I'll leave it to you to make this precise.