In this answer, in response to the question "How to find $2+7$ from $53$?", user @Cocopuffs provided a reference to a Jacobsthal sum, which enables obtaining a Gaussian integer, based on a given Gaussian integer prime norm:
There is a way described in Jacobsthal's "Über eine Darstellung der Primzahlen der Form $4n+1$", see [link][1], section $2$. Find any quadratic rest $r$ modulo your prime $p \equiv 1 (4)$; then, a choice for $a$ is $$a = \sum_{m=1}^{(p-1)/2} \Big(\frac{m}{p}\Big) \Big(\frac{m^2 + r}{p}\Big).$$ The above are Legendre symbols. For example, when we calculate this for $p = 53$, taking $r=1$, we get $a = 7$ - from this it is easy to find $53 = 7^2 + 2^2.$
Lipschitz quaternions are quaternions whose components are all integers: $$L=\{a+bi+cj+dk \in \mathbb{H} | a, b, c, d \in \mathbb{Z}\}$$
Do there exist any sums (Jacobsthal or otherwise), which enable obtaining a Lipschitz quaternion (or the full set of quaternions) corresponding to a given (prime or otherwise) quaternion norm?
More generally, are there any non-brute-force methods (that is enumerating all possible combinations of integers up to a given point and checking each sum individually) which produce the desired result (finding any or all Lipschitz quaternions, corresponding to a given norm or category of norms)?