Harold (mcintosh@servidor.unam.mx ) writes:

<<
An interesting intersection of these two trains of thought is the fact
that the Lie Algebra of O4 is the direct product of the Lie Algebras
of two O3 rotation groups
>>

And the rotation group SO(4) = S^3 x S^3 / (g,h) ~ (-g,-h)  (where S^3 is the group of unit quaternions).

Who would've thought that the rotation group SO(4) of something so round as the
3-sphere would be practically "square" (and its Lie algebra exactly square)?  There's a very nice discussion of this in the Thurston notes on 3D geometry & topology that were sold by MSRI before the corresponding book got published. I'm not sure whether this discussion is in the book, too.

--Dan