TITLE: How to visualize $SO(4) \simeq SO(3)\bigotimes SO(3) / \mathbb{Z}_2 $
QUESTION [6 upvotes]: Is there a simple way to see that $SO(4) \simeq SO(3)\bigotimes SO(3) /\mathbb{Z}_2$ ?

REPLY [3 votes]: Since $SO(3)$ is related to $SU(2)$ I recommend this paper :
More on the Isomorphism $SU(2)\otimes SU(2) \simeq SO(4)$ by Kazuyuki Fujii
, Hiroshi Oike and Tatsuo Suzuki.