Just in case anyone on math-fun is interested, I'm forwarding my response to an enquiry from Derek Smith :--- I've been doing some thinking about the matter raised in Conway & Smith (2003) sect.5.4; for what it's worth, my conclusions are the following. At first sight it might appear that the transformation from X = P Q to X = Q' P', with P,P' primes of norm p and Q,Q' primes of norm q, might prove amenable to some simple combinatorial manipulation of components. For example if X has i- and j-components equal, then (modulo unit migration) set P',Q' equal to P,Q with i- and j-components interchanged. Again, if p = q, then interchange the factors with P',Q' equal to Q,P so that the factorisation (unique modulo unit migration) remains unaltered. However consider a more general example: X has norm 323 = 17*19. Meta-commutation may change the component bag of neither factor: [3,2,2,0]*[1,3,3,0] = [1,3,3,0]*[3,2,2,0] = [-9,11,11,0] (commutative); of both: [3,-2,0,2]*[3,3,-1,0] = [4,1,1,1]*[4,0,0,1] = [15,5,3,8]; of norm-17 only: [3,2,2,0]*[3,3,1,0] = [0,3,3,-1]*[4,0,0,1] = [1,15,9,-4]; of norm-19 only: [3,-2,0,-2]*[0,1,3,3] = [4,1,1,1]*[3,2,2,0] = [8,9,13,3]; above X = s + p i + q j + r k is denoted [s,p,q,r] . Further numerical exploration soon suggests this conjecture: Given primes P,Q with unequal odd norms p,q, denote by S,T closure sets under inclusion of Y' in S and Z' in T, where Z' Y' = Y Z under meta-commutation, for Y,Z already in S,T; initially S = {P}, T = {Q}. Then (modulo unit migration) S,T finally comprehend _all_ Lipschitz integers with norm p,q resp. [By Jacobi's theorem, these sets have sizes p+1, q+1 resp.] It follows that any general meta-commutation algorithm may be expected to incorporate a mechanism enumerating representations of a rational integer as a sum of four squares, rendering any component-diddling solution highly improbable. The following refactorisation algorithm is almost certainly essentially optimal in general: set Q' = GCD_L(q, X) and P' = GCD_R(p, X), where GCD denotes the Euclidean algorithm applied appropriately, on left or right, to Lipschitz integers with at least one norm odd. Fred Lunnon