Jacquet-Langlands double coset decomposition

Question

Let $G=\mathrm{GL}(2,F)$ with $F$ a non-archimedean local field. Let $K=\mathrm{GL}(2,\mathcal{O}_F)$ be a maximal compact subgroup. Every element of $G$ lies in one of the double cosets \begin{equation*} KZ\left(\begin{array}{cc}1 & \\ & \varpi^m \end{array}\right)K \end{equation*} for $m\geq 0$ and $\varpi$ a uniformizer. In section 7 of Jacquet-Langlands, it is shown that these diagonal elements can actually be taken as representatives of a decomposition $T\backslash G/ K$, where $T$ is an elliptic torus. Does this have any generalization to $\mathrm{GL}(n)$ or other groups?