Let $G$ be a connected, reductive group over a number field $F$. Let $\gamma_1 \in G(F)$ be an elliptic element, i.e. one which is not a member of any proper parabolic subgroup of $G$. Let $\mathcal O$ be the conjugacy class of $\gamma_1$ as an element of the group $G(F)/Z(F)$. For $f \in C_c^{\infty}(G(\mathbb A)/Z(\mathbb A))$, define
$$K_{\mathcal O}(x,y) = \sum\limits_{\gamma \in \mathcal O} f(x^{-1}\gamma y)$$ which is a finite sum (I think). Then $K_{\mathcal O}(x,x)$ is integrable over $G(F) Z(\mathbb A) \backslash G(\mathbb A)$, with
$$\int\limits_{G(F) Z(\mathbb A) \backslash G(\mathbb A)} K_{\mathcal O}(x,x)dx = \operatorname{meas}(G(F)Z(\mathbb A) \backslash G(\mathbb A)) \int\limits_{G_{\gamma_1}(\mathbb A) \backslash G(\mathbb A)} f(x^{-1}\gamma_1x)dx$$
where $G_{\gamma_1}$ is the centralizer of $\gamma_1$. This is what Gelbart claims in Lectures on the Arthur-Selberg trace formula, page 9, Proposition 2.6. I'm trying to understand formally how this equality holds from the principle
$$\int\limits_{\Gamma_0 \backslash \Gamma_2} = \int\limits_{\Gamma_0 \backslash \Gamma_1} \int\limits_{\Gamma_1 \backslash \Gamma_2}$$ for subgroups $\Gamma_0 \subset \Gamma_1 \subset \Gamma_2$. I'm familiar with a similar computation in the case of a compact quotient, but I'm having trouble with the fact that the center is coming into play.
Gelbart says that the proof follows the same way as in the case of a compact quotient. I thought I would let $H = G(\mathbb A)/Z(\mathbb A)$, and $\Gamma = G(F)/Z(F) = G(F) Z(\mathbb A)/Z(\mathbb A)$. Then $\Gamma$ is a closed subgroup of $H$, with $\Gamma \backslash H = G(F) Z(\mathbb A) \backslash G(\mathbb A)$, so that
$$\int\limits_{G(F) Z(\mathbb A) \backslash G(\mathbb A)} K_{\mathcal O}(x,x)dx = \int\limits_{\Gamma \backslash H} K_{\mathcal O}(x,x)dx. \tag{1}$$ Now $\mathcal O$ is a conjugacy class in $\Gamma = G(F)/Z(F)$. Then
$$K_{\mathcal O}(x,x) = \sum\limits_{\Gamma_{\gamma_1} \backslash \Gamma} f(x^{-1}\gamma^{-1}\gamma_1 \gamma x).$$ My problem is that I don't know how to handle the centralizer $\Gamma_{\gamma_1}$ of $\gamma_1$ in the group $\Gamma = G(F)/Z(F)$. Is it true that $\Gamma_{\gamma_1} = G_{\gamma_1}(F)/Z(F)$? I doubt it. It's only clear that the right hand side is contained in the left.