If you want to see a constructive proof (modulo factoring the number into primes) that every positive integer is a sum of four squares, look here: Every positive integer is a sum of four integer squares. Note that the formula that could have been derived using quaternionic norms is just stated as such. I assume the formula itself is older than quaternions, although I don’t know that.
Still digesting your posts, but wanted to say thanks in the meantime. The Xconj(X) thing should have been obvious–my intuition was right, but I didn’t (quite) make the simple leap that since it always comes to a non-complex number, it must just be a scaled version of the other one (i.e., non-primitive). Or the degenerate case of a^2+0 in the case of Xconj(X)Yconj(Y).