Why should I expect the product of sum of four squares to be a sum of four squares? How did Euler come up with it?

Question

Euler discovered the lovely identity shown here:

https://en.wikipedia.org/wiki/Euler%27s_four-square_identity

Is there a natural reason to assume a solution can be found? Any intuition?

I saw that any symmetric bilinear form is diagonalizable, and so an intuition might be that if we get lucky when diagonalizing the basis won't contain complex numbers. However the product of 2 sums of 4 squares isn't a symmetric bilinear form, is there a more general diagonalizing statement that I'm not aware of to take care of those almost symmetric polynomials?

Answer 1

Citing this article by Winfried Scharlau from "E.A. Fellmann: Leonhard Euler 1707–1783: Beiträge zu Leben und Werk, Springer Verlag, 2013":

Parallel to his hitherto told research Euler always occupied himself with Fermat's theorems on the representation of numbers as sums of squares or polygonial numbers. His correspondence with Goldbach at this time gives a lively picture of his many and never dwindling attempts about these theorems. It must have been a close thought to him to try to apply the method of generating functions to Fermat's problems.

The letter to Goldbach, May 4th, 1748: Link

It starts at page 452 ("Folgendes theorema kann auch dienen in vielen Fällen die quatuor quadrata selbst zu bestimmen, woraus eine Zahl zusammen gesetzt ist")

The following theorem can in many cases serve to determine the quatuor quadrata of which a number is composed

(The language is German with a lot of Latin)

Another article by Herbert Pieper on Euler's attempts: Link