Lagrange’s Four-Square Theorem — a special case of the Fermat-Cauchy Polygonal Number Theorem (FCPNT) — states that every natural number can be written as the sum of the squares of at most four integers.
I’m trying to find a new proof of a conditional version of part of that theorem:
Theorem: If $n$ is a natural number such that $2n+1$ can be written as the sum of the squares of four integers, then $2n+1$ can be written as the sum of squares of four integers which sum to $1$. In other words, one can always find integers $t,u,v,w$ such that \begin{align} 2n+1 &= t^2+u^2+v^2+w^2, \\ 1 &= t+u+v+w. \end{align}
The non-conditional version of this has been proven many times in the past (cf. Pollock, Cauchy, Bradley, etc.); but as far as I know, it has always been proven by first assuming the Fermat-Gauss theorem [that every natural number is the sum of at most three triangular numbers].
I’m hoping to find an elementary proof — existing or new — of this conditional result that doesn’t rely on any other part of the FCPNT. Any references or hints would be appreciated.