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Given coprime positive integers $d',d''$, B\'ezout's Lemma tells us thatthere are integers $u,v$ so that $d'u-d''v=1$. We show that, interchanging $d'$and $d''$ if necessary, we may choose $u$ and $v$ to be Loeschian numbers,i.e., of the form $|\alpha|^2$, where $\alpha\in\mathbb{Z}[j]$, the ring ofintegers of the number field $\mathbb{Q}(j)$, where $j^2+j+1=0$. We do this byusing Atkin-Lehner elements in some quaternion algebras $\mathcal{H}$. We usethis fact to count the number of conjugacy classes of elements of order 3 in anorder $\mathcal{O}\subset\mathcal{H}$.

A refinement of B\'ezout's Lemma, and order 3 elements in some quaternion algebras over $\mathbb{Q}$