  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.
