Author’s summary: “The level \(s\) (resp. sublevel \(\underline{s}\)) of a ring \(R\) with \(1\neq 0\) is the smallest positive integer such that \(-1\) (resp. 0) can be written as a sum of \(s\) (resp. \(s+1\)) nonzero squares in \(R\), provided \(-1\) (resp. 0) is a sum of nonzero squares at all. D. W. Lewis showed that any value of type \(2^n\) or \(2^n +1\) can be realized as level of a quaternion division algebra, and in all these examples, the sublevel was \(2^n\), which prompted the question whether or not the level and sublevel of a quaternion division algebra will always differ at most by one. In this note, we give a positive answer to that question.”
The proof is by clever computation with (pure) quaternions, it is elementary. Note that it is still not fully known what exact values can be realized as (sub)levels of quaternion division algebras.