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It is mentioned that there is a subalgebra isomorphic to the alternatinggroup $2 \cdot A_4$ as a subalgebra of the Quaternion over integers andhalf-integers called Hurwitz quaternionic integers $\mathscr{H}$ in the book byJ.H.Conway and Neil J. A. Sloane. In this paper, I have followed this book andextended Quaternion over integers and half-integers to have duality, and provedthat a subalgebra in it isomorphic to Exceptional Weyl group $W(F_4)$. I havealso found a method of constructing the $16$-dimensional lattice $\Lambda_{16}$which seems to be isomorphic to the lattice called the Barnes-Wall lattice$\Lambda_{\text {Barnes-Wall }}$, which is currently considered to be verydense (although this remains to be discussed) using the Dual Quaternion.Lastly, I briefly mention how to construct an exceptional Weyl group $W(E_8)$using an Octonion and Dual Quaternion.

A construction of Exceptional Weyl Group $W\left(F_4\right)$ and  $W\left(E_8\right)$ using Quaternion, and the lattice in 16-dimensional  Euclidean space