Exploring Double Rotations in 4-Space: An Investigation in Group Theory

[zpconn]

I was wondering if anybody could help me understand the "double rotations" in 4-space. These are evidently rotations that fix only a single point--the center of rotation--and that take place in two hyperplanes simultaneously and independently.

Beyond that, I have an even more specific question. Suppose R1, R2, ..., Rn, where n is even, are reflections of 4-space in hyperplanes *through the origin*. Under what conditions is the product a "double rotation"? It's clear the result is a rotation of some kind: first, the products R1 * R2, R3 * R4, ... are each individually rotations; second, the product of two rotations fixing the origin will be another rotation fixing the origin (I don't think this is obvious since in 4-space the product of two rotations is not necessarily a rotation, but I've worked out a simple proof using quaternions).

 

[fresh_42]

The first question sounds like the direct product $SO(2)\times SO(2)$.
The second question is more complicated. Using quaternions seems to be a good idea. It is basically a word problem in some group, what you are asking for. Hence we must first try to determine the group. In general we have that the $3-$sphere can be viewed as $\mathbb{S}^3 \cong SO(4,\mathbb{R})/SO(3,\mathbb{R})\cong U(1,\mathbb{H})$.