Let ${\mathrm{M}}_2(\widehat{{\cal O}_K})$ be the $2 \times 2$ matrices over the finite adele of the full integer ring ${\cal O}_K$ of a totally real #-field $K$.
For the quaternion algebra $D_K$ over $K$, suppose that it is dense in the adelic matrices as follows$\colon$ \begin{equation*} D_K \overset{{\mathrm{dense}}}{\subset} {\mathrm{M}}_2(\widehat{{\cal O}_K} \otimes_{{\cal O}_K} K) \quad \cdots\cdots \quad (\lozenge). \end{equation*}
We assume that $D_K$ satisfies Eichler condition which ensures that $D_K$ splits at least one infinite place of $K$. By Eichler condition, it follows that all maximal ordres of $D_K$ is conjugate to each other up to an element of $D_K$. That is, $O_D = aO'_Da^{-1}$ by $a \in D_K$ for an arbitrary pair of maximal ordres $O_D, O'_D$ of $D_K$.
We fix one Eichler ordre ${\mathrm{M}}_2(N)$ of level $N$ inside ${\mathrm{M}}_2(\widehat{{\cal O}_K})$ as follows$\colon$ \begin{equation*} {\mathrm{M}}_2(N) \colon= \Bigg\{ \begin{pmatrix}\label{matrix} a & b \\ c & d \end{pmatrix} \subset {\mathrm{M}}_2(\widehat{{\cal O}_K}) ~|~ c \equiv 0 ~{\mathrm{mod}}~ N \Bigg\}. \end{equation*} For an arbitrary element $x \in ({\mathrm{M}}_2(\widehat{{\cal O}_K} \otimes_{{\cal O}_K} K))^{\times}$, we have two Eichler ordres in ${\mathrm{M}}_2(\widehat{{\cal O}_K} \otimes_{{\cal O}_K} K)$ as follows$\colon$ \begin{equation*} {\mathrm{M}}_2(N), x{\mathrm{M}}_2(N)x^{-1} \subset {\mathrm{M}}_2(\widehat{{\cal O}_K} \otimes_{{\cal O}_K} K) \quad \cdots\cdots \quad (\blacklozenge). \end{equation*}
By $(\lozenge)$ and $(\blacklozenge)$, we can consider two Eichler ordres $E \colon= D_K \cap {\mathrm{M}}_2(N)$ and $E' \colon= D_K \cap x{\mathrm{M}}_2(N)x^{-1}$ in $D_K$.