What are the orbits for this action of $\mathrm{SO}(2)$ on $S_3$?

Question

The group action defined by: $\Phi: SO(2) \times S_3 \rightarrow S_3$ such that: ($A, r$) $\rightarrow$ [$ \mathbb{1} \otimes A$]$\cdot$[$r$], where A is a standard matrix $2 \times 2$ of 2D rotations, $r$ is a point on $S_3$ such that $r_1^2+r_2^2+r_3^2+r_4^2=1$ and “$\cdot$“ means a normal multiplication.

My question is: what actually is an orbit of this action?

Answer 1

Hint Show that the given action of $SO(2)$ on $S^3$ is free. Thus, all of its orbits are embedded copies of $SO(2)$. Then, show directly that any orbit is contained within a $2$-dimensional subspace of $\Bbb R^4$.

Additional hint Hence the orbits are the intersections of $S^3$ with these planes, that is, they comprise a family of great circles in $S^3$. In fact, space of orbits is homemorphic to $S^2$, and the canonical quotient map $S^3 \to S^3 / SO(2) \cong S^2$ is the classical Hopf fibration. If you're already familiar with quaternions, it is an instructive exercise to identify $SO(2)$ with the space of unit complex numbers and $S^3$ with the space of unit quaternions and rewrite the action in terms of quaternion multiplication.