Gröbner-Shirshov bases and normal forms for some Coxeter groups

The Gröbner-Shirshov bases and normal forms of the infinite Coxeter groups of type $\widetilde{B}_n$, $\widetilde{C}_n$ and $\widetilde{D}_n$ are obtained for the first time. New versions of Gröbner-Shirshov bases and normal forms of the finite Coxeter groups of type $A_n$, $B_n$ and $D_n$ are also found. Using combinatorial techniques, the product of two normal forms is attained as normal form in all Coxeter groups mentioned above and the infinite Coxeter group of type $\widetilde{A}_n$. Hence all these groups are completely revealed.