I wonder if there exists a characterisation of Hurwitz integers which are represented as sums of two squares of Hurwitz integers, up to multiplication by a unit. And if so, could you please point to a reference?

By a Hurwitz integer I mean an integer in the ring of quaternions, that is, a quaternion whose components are either all integers or all half-integers.

It can be proved that we only need to concentrate on Hurwitz integers with integer components.

Thanks, and regards, Guillermo