It is also true that the automorphism group of the quadratic form is known. See http://mathoverflow.net/questions/110956/is-there-a-topograph-for-pythagorean-triples 
and the three matrices. If you have any particular $x^2 + y^2 - z^2 = n,$ write $(x,y,z)$ as a column vector. Multiply by any of the three square matrices or its inverse and you get another solution for $n.$  Multiply again you get another, and so on for any combination of group elements.