Proof regarding principal factors of the discriminant in $\mathbb{Q}(\sqrt{d})$

Question

So I understand there are (up to $\pm$) exactly two primitive (no rational integer factors) elements $\alpha_1 ,\alpha_2 \in \mathcal{O}_K$ such that the fundamental unit $\varepsilon$ of $K=\mathbb{Q}(\sqrt{d})$ is $$\varepsilon= \frac{{\alpha_1}^2}{N(\alpha_1)}=\frac{{\alpha_2}^2}{N(\alpha_2)}$$ In fact one can show that these are $\alpha_1=\frac{1+\varepsilon}{n}$ and $\alpha_2=\frac{1-\varepsilon}{m}$ where $m,n \in \mathbb{N}$ are chosen so that the elements are primitive. However I've often read by now that furthermore $$N(\alpha_1) \cdot N(\alpha_2)\in \{d,4d\}$$ with the second case occuring if and only if $d\equiv 3 \pmod 4$ and $2 \nmid N(\alpha_1)$. However I have not found a proof of this in any literature and fail to come up with one myself. The best I get is that $N(\alpha_1) \cdot N(\alpha_2)= k^2 \cdot d$ for some integer $k$ and I don't even see where and why the two cases are distinguished. Can anyone explain how to prove this fact or refer to literature where this is proven? Thanks ahead to anyone who takes their time out of their day to help me :)