Let $k$ a CM-field of conjugation $\bar \cdot$ and maximal totally real subfield $k^+$ and $k^{++}$ its positive part $k^{++}$. Given a totally positive element $x$ in the ring of integers of $k^+$, what are the arithmetic obstructions to write $x$ as an hermitian sum of two squares, that is as $$x = \alpha\bar\alpha+\beta\bar\beta?$$ (note that I want integers, not field elements, so I don't want to simply apply Siegel's four square theorem in $k^+$).
I guess this is the CM-analogue of Fermat's two square result (where for primes, we know that we only need to look at the mod 4 valuation / Legendre symbol at -1 to conclude). Analogously I suppose we then want the ideal $x\mathcal{O}_{k^+}$ to split in $k$, but beyond this I can't prove the the decomposition exists.
