Nice observations! In fact, I've seen numerous claims that the almost-squareness of SO(4) is the reason that 4-dimensional geometry is supposedly so strange. But as much as it's clear that 4-dimensional *topology* is extremely strange (e.g., the uncountably many inequivalent smooth structures on R^4 as compared with only 1 on every other R^n), I'm not that familiar with anomalies in 4D *geometry*. Other than the three regular polytopes not found in other dimensions, especially the 24-cell. --Dan On Jul 28, 2014, at 5:44 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Indeed, that's very interesting.
Observe that Spin(4) = Spin(3) x Spin(3) has an obvious centre of order 4 (namely {(1,1), (1,-1), (-1,1), (-1,-1)}). Every subgroup of this is normal (that's how centres work), so we get five nice normal subgroups. Considering their quotients, we have:
Subgroup: {(1,1)} Quotient: Spin(4) = Spin(3) x Spin(3)
Subgroup: {(1,1), (-1,1)} Quotient: Spin(3) x SO(3)
Subgroup: {(1,1), (1,-1)} Quotient: SO(3) x Spin(3)
Subgroup: {(1,1), (-1,-1)} Quotient: SO(4)
Subgroup: {(1,1), (1,-1), (-1,1), (-1,-1)} Quotient: PSO(4) = SO(3) x SO(3)
Consequently, SO(4) enjoys the status of being the double cover of SO(3) x SO(3), and having Spin(3) x Spin(3) as its own double cover! It seems really weird for a group like SO(4) to be neatly sandwiched between two direct squares in this way.