This is a follow-up of my recent question on real values of the $j$-invariant. It has had a partial response so far, establishing that any "reality root" $n$ is indeed rational. Now it appears that this can be taken a step further, somewhat in the inverse direction:
In fact, I have noticed that seemingly for $a,b,n\in\mathbb N$, the algebraic value $j\left(\dfrac {a+\sqrt{-n }}{b}\right)$ always has at least one real Galois conjugate (i.e. among the roots of its minimal polynomial), which can moreover always be written as one of the following three forms (for a certain $m\in\mathbb N$): $$j\left( {\sqrt{-m }} \right),\ j\left(\dfrac {1+\sqrt{-m }}{2}\right),\ \text { or }\ j\left(\dfrac {1+\sqrt{1-m^2 }}{m}\right).$$
Is this already stated anywhere in the literature? If not, how to prove it?
(Or are there possibly cases I have missed?)
