My C program looks at a billion or so products of elementary rotations in 4D for those close to the identity.
Rij(θ) is the rotation of θ radians in the i-j plane (i≠j).
Rij(θ)−1 = Rji(θ) = Rij(−θ).
The search excludes trivial identities such as AA−1 = I and
R12(θ)R34(θ)R21(θ)R43(θ) = I.
It found 144 instances of products of 8 rotations which were the identity rotation to 16 significant digits.
Typical of the products are:
R21(θ)R32(θ')R10(θ')R30(θ)R12(θ)R23(θ')R01(θ')R03(θ). In 3D such generators produce a free group for most values of θ. This is germane to the paradoxical duplication of the 3D unit ball by Tarski-Banach. Indeed all of the identities involved all 4 coordinates.
This discovery was by calculating with orthogonal matrices. The Clifford calculation corroborates the coincidence by appending the following to these tools.
(define mull (lambda lst (let m ((l lst))(if (null? l) C1 (C* (car l) (m (cdr l))))))) (define (d0 a)(C+ (sm (cos 0.5) C1)(sm (sin 0.5) a))) (define (d1 a)(C+ (sm (cos 0.6) C1)(sm (sin 0.6) a))) (mull (d0 (C* g2 g1)) (d1 (C* g3 g2)) (d1 (C* g1 g0)) (d0 (C* g3 g0)) (d0 (C* g1 g2)) (d1 (C* g2 g3)) (d1 (C* g0 g1)) (d0 (C* g0 g3))) ; => nearly one.This suggests, but does not prove, that there is some n(n−1)/2 (not linear) subspace of Cn that is closed under multiplication and forms a double cover of the orientations. This is presumably a Lie group and is generated by the unit vectors. I would like very much to know the dimensionality of this group. Perhaps I can work out the structure coefficients of the Lie algebra.
Even if this is so there may be other Clifford numbers in the Clifford group that are not products of unit vectors.
The C program also discovered A product that is somewhat near the identity. Its corresponding Clifford number is ni below:
(define (mx fac)(if (null? fac) C1 (C* (d0 (car fac)) (mx (cdr fac))))) (define ni (mx (list (C* g0 g1)(C* g3 g1)(C* g3 g2)(C* g3 g0)(C* g2 g0) (C* g1 g0)(C* g1 g3)(C* g2 g3)(C* g3 g0)(C* g0 g2)))) ni ; => ((((0.99999798 . 0) . (0 . 0.00114777)) . ((0 . -0.00033676) . (-0.00096580 . 0))) . (((0 . -0.00095320) . (-0.00062014 . 0)) . ((-0.00062014 . 0) . (0 . -3.3447e-17))))This corresponds to the orthogonal matrix computed by the C program:
0.999997 0.001238 0.001244 0.001905 -0.001242 0.999997 0.001932 0.000670 -0.001237 -0.001932 0.999995 -0.002297 -0.001907 -0.000677 0.002294 0.999995 (Om ni) ; => ((0.99999664 0.00123843 0.00124365 0.00190538) (-0.00124211 0.99999714 0.00193161 0.00067012) (-0.00123688 -0.00193160 0.99999473 -0.00229734) (-0.00190740 -0.00067692 0.00229368 0.99999532))which is good agreement.