Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$

Suppose $n$ is the fundamental discriminant associated with a quadratic extension of $\mathbb{Q}$. We show that for every Diophantine $m$-tuple $\{t_1, t_2, \ldots, t_m\}$ with the property $D(n)$, there exists integral ideals $\mathfrak{t}_1, \mathfrak{t}_2, \ldots, \mathfrak{t}_m$ of $\mathbb{Q}(\sqrt{n})$ and $c\in \{1,2\}$ such that $t_i= c\mathcal{N}(\mathfrak{t}_i)$ for $i=1,2, \ldots, m$. Here, $\mathcal{N}(\cdot)$ denotes the norm map from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}$. Moreover, we explicitly construct the above ideals for Diophantine pairs $\{a_1, a_2\}$ whenever $\gcd(a_1, a_2) = 1$.