Is there a reference for a proof that $\text{SO}(4)$ is homeomorphic to $\text{SO}(3)\times S^3$? Since $\text{SO}(4)$ acts transitively on $S^3$ with stabilizer $\text{SO}(3)$, we have a fiber bundle $\text{SO}(3)\to \text{SO}(4)\to S^3$, but I can't see why this bundle is trivial.
2026-03-30 17:02:03.1774890123
$\text{SO}(4)$ is homeomorphic to $\text{SO}(3)\times S^3$
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Here is a reference about the topology of SO(4):
http://www.gregegan.net/ORTHOGONAL/03/WavesExtra.html#topology
You can also see this article of wikipedia:
https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space#Group_structure_of_SO(4)
Which describes precisely the universal cover of SO(4).