  In this paper, we review the properties and representations of the Weyl
groups relevant in the study of discrete integrable systems. Previously in
\cite{jns4, Shi:19}, properties of Weyl groups of type $ADE$ (known as
simply-laced) were shown to be useful in characterizing and establishing
relations between different integrable systems. Here we extend the formulations
and discussions to include non-simply-laced types, giving special attention to
developing formulas related to the translational elements of the affine Weyl
groups. As applications, we show how these are used to clarify the natures of
some integrable systems of type $E_8$ \cite{NN_elliptic} and $F_4$
\cite{ahjn:16} appeared recently in the literature.
