On the automorphism of Barns Wall Lattice ${\mathrm{\Lambda }}_{B{W}_{16}}$ and rank 4 tensor of quaternions

In a previous paper, I found that the Weyl group $W(F_4)$ and Barns-Wall Lattice $B{W}_{16}$ 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 ${\mathcal{H}}^4$, isomorphic to the automorphism $\text{Aut}(B{W}_{16})$ order $2^{21}\cdot 3^5\cdot 5^2\cdot 7$ of Barns Wall Lattice $B{W}_{16}$ by functionally extending the rank of the tensor product of quaternions to $4$.