Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup of $G$ in good position relative to $P_0$, so we have $G = P_0K$ (if $G$ is unramified, we can take $K$ to be a conjugate of $G(\mathcal O_k)$).

For each standard parabolic $P$, let $M$ be the unique Levi subgroup of $P$ containing $A_0$.

Does there always exist an open subgroup $H$ of $K$ such that we have $$H = \bigg(\prod\limits_{\alpha \in \Phi(A_M,P)_{\textrm{red}}}(H \cap N_{-\alpha})\bigg)(H \cap M)\bigg(\prod\limits_{\alpha \in \Phi(A_M,P)_{\textrm{red}}}(H \cap N_{\alpha})\bigg)$$? Here $A_M$ is the split component of $M$, and $N_{\alpha}$ is the product of the root subgroups $U_a$ for $a \in \Phi(A_0,G), a|_{A_M} = \alpha$.

This is claimed in Waldspurger's notes on the Plancherel measure for $p$-adic groups, based on Harish-Chandra's lectures, Section I.1.

No proof was given for the existence of $H$. I was wondering if anyone knew a reference for this result.