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

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 $\mathcal{H}$ 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 ${\mathrm{\Lambda }}_{16}$ which seems to be isomorphic to the lattice called the Barnes-Wall lattice ${\mathrm{\Lambda }}_{\text{Barnes-Wall }}$, 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.