Gr\"{o}bner-Shirshov Bases and Normal Forms for The Infinite Coxeter Groups of Types $\widetilde{B}_{n}$ and $\widetilde{D}_n$

In this paper, we will investigate the infinite Coxeter groups $\widetilde{B}_n$ and $\widetilde{D}_n$. Their Gr"{o}bner-Shirshov bases and classifications of normal forms are achieved by leveraging results from the infinite Coxeter groups of types $\widetilde{C}_n$. Additionally, a new algorithm is presented for obtaining normal forms of elements within these groups.