is the scalar part of P Q', which is zero if the quaternions are orthogonal, i.e., P Q' + Q P' = 0 if P,Q are orthogonal. We now prove that 1-ab is orthogonal to a+b: (a+b)(1-ab)'+(1-ab)(a+b)' = (a+b)(1-b'a')+(1-ab)(-a-b) = (a+b)(1-ba)+(ab-1)(a+b) = a+b-aba-bba+aba+abb-a-b = -bba+abb = a-a = 0 Similarly, 1+ab is orthogonal to a+b, 1+ab is orthogonal to a-b, and 1-ab is orthogonal to a-b. (a+b)(1+ab)'+(1+ab)(a+b)' = (a+b)(1+b'a')+(1+ab)(-a-b) = (a+b)(1+ba)+(1+ab)(-a-b) = a+b+aba+bba-a-b-aba-abb = bba-abb = -a+a = 0 (a-b)(1-ab)'+(1-ab)(a-b)' = (a-b)(1-ba)+(1-ab)(-a+b) = a-b-aba+bba-a+b+aba-abb = bba-abb = -a+a = 0 (a-b)(1+ab)'+(1+ab)(a-b)' = (a-b)(1+ba)+(1+ab)(-a+b) = a-b+aba-bba-a+b-aba+abb = -bba+abb = a-a = 0 Therefore, we have shown that the map v |-> AvB can be factored into the composition of the map v |-> A1 v B1 and the map v |-> A2 v B2. Furthermore, the first map rotates about the 'axis' plane defined by the points 0, a-b, 1+ab, while the second map rotates about the 'axis' plane defined by the points 0, a+b, 1-ab, and the two 'axis' planes are orthogonal to one another. The pure vectors a,b may be computed from A,B as follows: A = cos(gamma) + a sin(gamma), so Re(A) = (A+A')/2, and a = (A-Re(A))/sqrt(1-Re(A)^2); similarly, Re(B) = (B+B')/2, and b = (B-Re(B))/sqrt(1-Re(B)^2). Conclusions ----------- Given an arbitrary 4x4 special orthogonal matrix O (det(O)=1), we have shown two different ways to construct unit quaternions A,B from O, such that the action vO of the matrix O on the 4-vector v=[w,x,y,z] is the same as the quaternion triple product A Q(w,x,y,z) B. Furthermore, we have shown how to factor this triple product into the product A1 A2 Q(w,x,y,z) B2 B1, so that the inner mapping fixes one 'axis plane' and the outer mapping fixes an orthogonal 'axis plane'. Finally, we have shown how to compute 3 points on each of these axis planes. References ---------- Coxeter, H.S.M. Regular Complex Polytopes, 2nd Ed. Cambridge University Press, Cambridge, 1991. ISBN 0-521-39490-2. (Especially Chapter 6: "The geometry of quaternions".) Eves, Howard. Elementary Matrix Theory. Dover Publications, Inc., New York, 1966. ISBN 0-486-63946-0. Neumann, Peter M., Stoy, Gabrielle A., and Thompson, Edward C. Groups and Geometry. Oxford University Press, 1994. ISBN 0-19-853451-5. Especially Chapter 16: "Complex numbers and quaternions". Salamin, Eugene. "Application of Quaternions to Computation with Rotations". Stanford AI Lab. Internal Working Paper, 1979. On the World-Wide Web at ftp://ftp.netcom.com/pub/hb/hbaker/quaternion/stanfordaiwp79-salamin.ps.gz (also .dvi.gz).