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$$
\def\mod{\mathrel{\big/}}
\Dg
{\rm Spin}(3) &        &       &       & SU(2)       \\
              & \rdTo  &       & \ldTo               \\
              &        & SO(3)                       \\
              &        & \dTo                        \\
& O(3)\mod O(1)^3 = {} & SO(3)\mod(O(1)^3\cap SO(3)) \\
\endDg
$$