is standard terminology in abstract building theory. This choice is made to avoid confusion: alcoves here are chambers (in the abstract sense) in Euclidean buildings and parallelism classes of Weyl chambers here are chambers (in the abstract sense) in spherical buildings at infinity of Euclidean buildings [@WeissAffine Chap. 8], [@AbramenkoBrown 11.8].

It is easy to see that, in order to prove that the map $d$ defined thanks to axioms [(EB 1)-(EB 3)]{} is a distance, it remains to check that the triangle inequality holds; this is mainly done by using the retraction given by axiom [(EB 5)]{}. The previously quoted metric motivation (Remark \[rk - metric motivation\]) so to speak became a definition. Note that the existence of suitable retractions is useful to other purposes.

The following examples of possibly non-simplicial Euclidean buildings correspond to the examples of simplicial ones given in Example \[ex - simplicial buildings\].

\[ex - non-simplicial buildings\]

-   Consider the real line $\Sigma = {\bf R}$ and its isometry group ${\bf Z}/2{\bf Z} \ltimes {\bf R}$. Then a Euclidean building of type $({\bf R}, {\bf Z}/2{\bf Z} \ltimes {\bf R})$ is a real tree – see below.

-   For a $2$-dimensional case extending simplicial $\widetilde{{\rm A}_2}$-buildings, 