The following is a well-known result.
Theorem: If $t$, $u$, $v$, and $w$ are coprime integers, with $u$ odd, satisfying \begin{align} t^2 &= u^2 + v^2 + w^2, \end{align} then there exist non-negative integers $a$, $b$, $c$, and $d$, with $a+b+c+d$ odd, such that \begin{align} t &= a^2 + b^2 + c^2 + d^2, \\[0.2em] u &= a^2 + b^2 - c^2 - d^2, \\[0.2em] v &= 2(ac+bd), \\[0.2em] w &= 2(ad-bc). \end{align}
It’s variously attributed to Catalan, Lebesgue, etc.; the first proof considered “sufficient” is credited to Dickson.
I have two questions:
What is the most elementary proof that is still considered sufficiently rigorous? Carmichael gives one in his Diophantine Analysis (§10) that would have been available to Fermat… but apparently it’s not considered "sufficiently rigorous" (?).
Is there a version in which $a,b,c,d$ are all positive [if $t,u,v,w$ are all positive]?