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Aug 20, 2016 at 12:12 comment added Clément de Seguins Pazzis Thank you very much. On the significance of the determinant being a square: if the ring is a field and the determinant is nonzero then the matrix under consideration turns out to represent the polar form $(x,y) \mapsto N(x+y)-N(x)-N(y)$ of the norm $N$ of some quaternion algebra. This can be used to give an alternative proof of the formula but it requires a lot of additional background.
Aug 20, 2016 at 11:53 vote accept Clément de Seguins Pazzis
Aug 20, 2016 at 11:09 history answered Colin McQuillan CC BY-SA 3.0