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?