I read recently that Gauss provided a proof of Lagrange's Four-Square Theorem using his ideas about equivalence classes of quadratic forms (i.e. linear substitution of variables) somehow applied to $w^2+x^2+y^2+z^2$ but I can't find any mention of this in online literature.

Can anyone show me the proof or does it not exist?

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    $\begingroup$ Gauss proved the three-square theorem using binary quadratic forms, see here. Then he pointed out that this can be used also for the four-square theorem - see this question. If you google for the three-square theorem you'll find more than enough online literature. $\endgroup$ – Dietrich Burde Mar 20 at 15:37

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