Prove that the group $H^*$ is Isomorphic to the group $S^3 \times R$?

Question

I'm trying to prove that the quaternion group $H^*$ is isomorphic to the direct product $S^3\times R^+$ where $S^3$ is the 3-sphere which has unit length 1. And $R^+$ being the group of positive real numbers.

We know that there's a relationship which exists between the groups in which that $\{x=R, x>0\}$ but I do not know how to carry on any further.

Could someone help?

Answer 1

Hint: the 3-sphere is the set of unit vectors in Euclidean 4-space and the quaternions (including 0) are a model of 4-space.

Answer 2

Hint:

Any quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$ can be written in the form: $z=|z|\mathbf{u}$, where $|z|=\sqrt{a^2+b^2+c^2+d^2}\in \mathbb{R}^+$ and $\mathbf{u}=\frac{a}{|z|}+\frac{b}{|z|}\mathbf{i}+\frac{c}{|z|}\mathbf{j}+\frac{d}{|z|}\mathbf{k}$ is a unitary quaternion that can be see as a vector on the 3-sphere of radius $1$ since its components satisfy: $$ \left(\frac{a}{|z|}\right)^2+\left(\frac{b}{|z|}\right)^2+\left(\frac{c}{|z|}\right) ^2+\left(\frac{d}{|z|}\right)^2=1 $$ .