Maximal parahoric subgroups I am trying to prove Proposition 2.7.3 from I.G. Macdonald's book "Spherical Functions on a Group of $p$-adic Type":


Proposition 2.7.3: Every compact subgroup of $G$ is contained in a maximal compact subgroup. The maximal compact subgroups are precisely the maximal parahoric subgroups, and form $(l+1)$ conjugacy classes. All parahoric subgroups are open and compact. $G$ itself is locally compact, but not compact.


For context, here $G$ is a simply connected group of $p$-adic type (and specifically with a specific $BN$-pair and subgroups $(U_\alpha)_{\alpha\in \Sigma}$ defined earlier in the book, $N$ and $U_\alpha$ are closed subgroups of $G$ and $B$ is an open compact subgroup of $G$).
The second half of this seems quite straightforward (and it mentions it in the book), where the parahoric subgroups are finite unions of cosets $BwB$ so they are compact, while $G$ is not a finite union of the $BwB$ so it isn't compact.
But I can't quite wrap my head around the first part.
Thank you in advance!
 A: A geometric approach, avoiding BN pairs, is more informative and intuitive: 
There exists a certain $l$-dimensional simplicial complex (a Bruhat-Tits building) $X$ on which $G$ acts via automorphisms, transitively on chambers (maximal simplices). Fixing one such chamber (a fundamental chamber) one defines maximal standard parahoric subgroups of $G$ as stabilizers of the vertices of this chamber: Since the chamber has dimension $l$, it has $l+1$ vertices. These vertices have disjoint orbits, hence, there are $l+1$ conjugacy classes of parahoric subgroups of $G$, represented by the standard parahorics.  
Compactness of parahorics is a consequence of the Arzela-Ascoli theorem. 
The complex $X$ has nonpositive curvature, which implies the existence and uniqueness of a center of mass of any bounded subset. In particular, since orbits of compact subgroups $H< G$ are bounded, the center of such an orbit is a point $x$ fixed by $H$. Since the action is simplicial, $H$ will fix pointwise the smallest face of $X$ containing $x$. By conjugating $H$ in $H$, this face is a face of the fundamental chamber. Hence, $H$ fixes one of the vertices of the fundamental chamber, hence, is a subgroup of one of the standard parahoric subgroups.  
