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Show that SO(4) is isomorphic to the quotient of SU(2) X SU(2) by the subgroup generated by (-1,1)

Show that SO(4) is isomorphic to the quotient of SU(2) X SU(2) by the subgroup generated by (-1,1).

Solution Preview

Thinking of the quaternions |H as R^4, an arbitrary
element of SO(4) can be written (as a linear transformation)
v -> g_1 v g_2^{-1} where g_1, g_2 are unit quaternions.

this gives a double covering SU(2) x SU(2) --> SO(4)

details: the unit ...

Solution Summary

It is shown that SO(4) is isomorphic to the quotient of SU(2) X SU(2) by the subgroup generated by (-1,1).

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