On a special loop, the discon form, and the lattice connected with $O_7(3)$

A special commutative Moufang loop $\mathbf H$ of order $3^4$ is described. With the help of this loop, a trilinear Dickson form is constructed whose automorphism group is a Chevalley group of type $E_6$. Next, with the help of $\mathbf H$, a 27-dimensional representation is constructed for $O_7(3)$ over $\mathbf Z[\zeta]$, $\zeta^3=1$. This makes it possible to prove anew the embedding $O_7(3)\subseteq{}^2E_6(2)$. A similar construction concerning the embedding $L_4(3)\subseteq F_4(2)$ is described.