Let $A$ be a Coxeter complex which is euclidean, so I assume that $A$ is an affine space over the reals on which a Coxeter group $(W,S)$ acts, the elements of $S$ are reflections and I assume the complex to be simplicial, i.e., each chamber is a simplex.
So let $C$ be a chamber with vertices $v_0,\dots,v_d$.
Choosing $v_0$ as origin makes $A$ a euclidean space and $v_1,\dots,v_d$ is a basis of that space.
I am interested in the lattice
$$
\Lambda=\bigoplus_{j=1}^d {\mathbb Z}v_j.
$$
Is it true that $\Lambda$ contains all vertices of the complex? Does it even coincide with the set of vertices of the complex?