It is mentioned that there is a subalgebra isomorphic to the alternating group $2 \cdot A_4$ as a subalgebra of the Quaternion over integers and half-integers called Hurwitz quaternionic integers $\mathscr{ H}$ in the book by J.H.Conway and Neil J. A. Sloane. In this paper, I have followed this book and extended Quaternion over integers and half-integers to have duality, and proved that a subalgebra in it isomorphic to Exceptional Weyl group $W(F_4)$. I have also 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 very dense (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.
Categories: math.GM hep-th