Some more comments: On Jun 26, 2014, at 10:04 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Theorem 5.1. The reflection in the hyperplane sum(y_nu x_nu)=0 is represented by the transformation x -> -y conj(x) y. The product of two such reflections ... is a rotation.
The product of any even number of any kind of reflections in any Euclidean space is a rotation.
Theorem 5.2. The general rotation through angle phi (about a plane [this is 4D, remember]) is x -> a x b, where N(a)=N(b)=1 and S(a)=S(b)=cos(phi/2). Conversely, [every] transformation x -> a x b (N(a)=N(b)=1) is a rotation whenever S(a)=S(b).
Given two arbitrary fixed unit quaternions p and q, any mapping of R^4 = H to itself given by f_p,q(x) := pxq is a rotation of 4-space. The induced mapping h: S^3 x S^3 -> SO(4) via h(p,q) := f_p,q is a surjective homomorphism, with kernel only {(1,1),(-1,-1)} == Z_2. And so SO(4) == S^3 x S^3 / Z_2 as Lie groups. It's strange that something as symmetric as a rotation group should be that close to a cartesian square (or vice versa). (As in any Euclidean space, the general rotation fixes each of a maximal orthogonal set of 2-planes -- two of them in the case of R^4 -- each with its own rotation angle.) A left multiplication L_q(x) = qx alone corresponds to rotating all the vectors of the unit sphere S^3 by the same angle along the circles of a geometric Hopf fibration of one handedness (the original kind); a right multiplication R_p(x) = xp alone ditto, of the opposite handedness. In either case (and only these) is the orthogonal set of 2-planes non-unique. --Dan