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We present the manifestly covariant canonical operator formalism of a Weyl invariant (or equivalently, a locally scale invariant) gravity whose classical action consists of the well-known conformal gravity and Weyl invariant scalar-tensor gravity, on the basis of the Becchi-Rouet-Stora-Tyupin (BRST) formalism. It is shown that there exists a Poincar${\rm{\acute{e}}}$-like $\mathit{IOSp}(8|8)$ global symmetry as in Einstein's general relativity, which should be contrasted to the case of only the Weyl invariant scalar-tensor gravity where we have a more extended Poincar${\rm{\acute{e}}}$-like $\mathit{IOSp}(10|10)$ global symmetry. This reduction of the global symmetry is attributed to the presence of the St\"{u}ckelberg symmetry., Comment: 34 pages

In a previous paper, I found that the Weyl group $W(F_4)$ and Barns-Wall Lattice $BW_{16}$ can be constructed using the rank $2$ tensor of the quaternion. In the present paper, I describe how I were able to construct an algebra, which is the subalgebra of the direct product of Hurwitz Quaternionic integers $\mathscr{H}^4$, isomorphic to the automorphism $\text{Aut}(BW_{16})$ order $2^{21} \cdot 3^5 \cdot 5^2 \cdot 7$ of Barns Wall Lattice $BW_{16}$ by functionally extending the rank of the tensor product of quaternions to $4$.

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.