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Jun 14, 2015 at 17:22 comment added Will Jagy Annals of Mathematics, volume 41 (1940) pages 754-766, On the Rational Automorphs of $x_1^2 + x_2^2 + x_3^2.$ I suggest you also get Lam, ams.org/bookstore-getitem/item=gsm-67 since i think that part of your question is obstructions with coefficients in the field of rational functions $\mathbb F(x_1,\ldots, x_n)$ see also Rajwade, references at en.wikipedia.org/wiki/Hurwitz_problem
Jun 14, 2015 at 16:37 comment added A.T.Saaki Thank you! I made an awful mistake here for odd cases. Could you point out some references about Gordon Pall's work?
Jun 14, 2015 at 4:47 history answered Will Jagy CC BY-SA 3.0