Infinite families of Diophantine quadruples in $\mathbb{Z}[\sqrt{-2}]$ in the remaining exceptional congruence classes
We continue the study of $D(z)$-quadruples in the ring $\mathbb{Z}[\sqrt{-2}]$. Motivated by the earlier classification due to the authors and by the subsequent partial results for the remaining families, we consider the exceptional congruence classes arising in the forms $24a+5+(12b+6)\sqrt{-2}$, $24a+2+(12b+6)\sqrt{-2}$, and $48a+44+(24b+12)\sqrt{-2}$. By combining the regular extension method with new families obtained by fixing a divisor $e\mid 3z$ and a small element $v\in \mathbb{Z}[\sqrt{-2}]$, we construct explicit $D(z)$-quadruples in each of the previously unsolved congruence classes. More precisely, we show that every exceptional class contains infinitely many values of $z$ admitting a twice semi-regular $D(z)$-quadruple, i.e., a quadruple containing two regular $D(z)$-triples. We also include remarks on the exceptional values $z\in\{-1,1\pm 2\sqrt{-2}\}$ and on a computational search in the exceptional congruence classes.