Let be a Coxeter group where is generated by elements. Also by for we mean the minimal length is a product decomposition of Let us first use the book definition.
Definition 1. A Weyl distance function is a map where a set whose elements are called chambers such that
- if and such that then If we also have that then
- if then for any we have a chamber such that and
The pair is called a – metric space. The triple is called a building.
I’m fairly certain I copied correctly (triple checked), but it looks like or the first sentence in at least. Of importance is the fact we previously mentioned: that chambers of a Coxeter complex coincide with elements of the Coxeter group (since they are (standard) cosets of the trivial standard subgroup). If we thus let then can be thought of as a product on By condition the element is thus not an identity with respect to this product. Also in this regard one can show that the chambers of a Coxeter complex form a building where
with being the gallery metric we previously defined.
Conversely we can say two elements in are -adjacent if and -equivalent if they are either -adjacent or equal. If and are -equivalent, we write This is an equivalence relation since
Proposition 2. if one takes a generator value.
Proof. Suppose By part of the definition there is a such that and Thus by we have and thus
The equivalence classes under are called -panels. A panel is an -panel for some Galleries can be defined similarly with this terminology. Thus
 Abramenko, Peter and Kenneth Brown. Buildings. Graduate Texts in Mathematics. Vol. 248. Springer Science and Business Media. 2008.