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How to visualize the double cover of the rotational symmetry group of $S^3$ (which is $Spin(4)$, namely the double cover of $SO(4)$) as two copies of $S^3$?

This is due to $Spin(4) = SU(2) \times SU(2) = S^3 \times S^3$. But how to visualize the above via the perspective of $S^3$?

Related question: Implications of $\text{Spin}(3)\times \text{Spin}(3) \cong \text{Spin}(4)$

annie marie cœur
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  • You may enjoy Greg Egan's SO(4) applet – PM 2Ring Apr 29 '21 at 03:32
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    It's debatable how much about this situation can be visualized. You can show all rotations are a product of a left and a right isoclinic rotation, and hence such pairs forms a double cover - which turns out to be Spin(4) - by making a simple observation about the parametrization $(\alpha+\beta,\alpha-\beta)$. You can argue any (strictly) left/right isoclinic rotation has a pair of invariant planes containing any given vector by arguing about $\Bbb R^4$ being made into a complex vector space, and then use that to see pairs of isoclinic rotations form an $S^3\times S^3$. This is mostly algebra. – anon Apr 29 '21 at 03:45

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