By considering $S^3$ as the group of unit quaternions, and letting it act on itself from both the left and right, one can get an isomorphism $SO(4)\cong (S^3\times S^3)/C_2$, where the $C_2$ subgroup is generated by $(-1,-1)$. This means $SO(4)$ is a two-sheeted covering of $SO(3)\times SO(3)$. Is there a way to see this covering directly from the exceptional Lie algebra isomorphism $D_2\cong B_1\times B_1$?
2026-03-30 13:39:08.1774877948
Isometries of S^3 and some Lie algebras
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