  We consider the Cayley graph ${\rm C}(W,S)$ of a Coxeter system $(W,S)$ and
describe all maximal $2$-cliques in this graph, i.e. maximal subsets in the
vertex set such that the distance between any two distinct elements is equal to
$2$. As an application, we show that every automorphism of the half of Cayley
graph is uniquely extendable to an automorphism of the Cayley graph if $|S|\ge
5$.

Combinatorics;Group Theory