direction tangent bundle of $S^2$ homeomorphism

Question

Let $\triangle S^{2}$ ( the tangent direction bundle of $S^2$) be the unit tangent bungle of $S^2$ where the opposite tangent directions are identified.
Let G be the (order four) group of quaternions generated by $i$. Let us identify the unit quaternions with $S^3$.
Find an homeomorphism between $\triangle S^{2}$ ans $S^3/G$ .
My best bet was trying to use the Hopf fibration to get a map from $S^3$ to $S^2$, and try to use the tangent direction to get a point in the inverse image of the hopf fibration up to $i$ , but even if both the tangent direction and the fiber of the hopf fibration are homeomorphoc to $S^1$, I was unable to find a way to link them continuously.