Consider the tome of Bruhat and Tits: Groupes réductifs sur un corps local : I. Données radicielles valuées. Publications Mathématiques de l'IHÉS, 41 (1972), p. 5-251. (available on [NUMDAM](http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1972__41__5_0)). I am interested primarily in the statements of Propositions 4.4.3 and 4.4.4 (and maybe also 7.3.1).


In the swathe of notation and technical conditions present, I find it hard to read exactly what the precise statements of these two propositions are. My question is, can anyone give


(a) a precise version of the statements of 4.4.3 and 4.4.4?


(b) if the request in (a) is too much, some sort of simplified version that is easy to state/comprehend and hopefully still reasonably general.


or


(c) an alternative reference covering at least the case of a split reductive group?


(this question is related to, but more general than [Dinakar's question](https://mathoverflow.net/questions/2229/iwasawa-decomposition))



Iwasawa and Cartan Decompositions.
I think the key point is the proposition 4.4.2, where "good" subgroups are caracterised geometrically as stabilisers of special subgroups (ie, stabilisers of a point o such that the Weyl group W is the semidirect product of its translations and of the stabiliser of o in W).


Then the group G is the product of B (the stabiliser of a class of sector, a minimal parabolic group for an algebraic group) and the group K. Moreover, the group B itself is the product of B^0 (which is the union of pointwise fixators of sectors) and the group of translations (acting on some apartment containing o and a sector in this class). 


The Cartan decomposition is, as usual, the decomposition of an element g in kvk', where k and k' are element of K and v is an element which sends o to a vertice of the sector starting at o in the class defined by B.


The proposition 4.4.4 is meant to explain the relation between the two decompositions (ie, when you know the translation part in the Iwasawa decomposition, can you deduce it in the Cartan decomposition ?)


If you know how to attach a building to a reductive group, then the book "Buildings" by Abramenko and Brown is a good reference (see chap. 11), much easier to read. They treat every building, but construct only the affine building associated to SL(n). Another reference is the small book of Macdonald, "Spherical functions on a group of p-adic type" (chapter II, Theorem 2.6.11)


