Geodesic
On a~special loop, the discon form, and the lattice connected with~$O_7(3)$
Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 145-167

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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. 
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     author = {V. P. Burichenko},
     title = {On a~special loop, the discon form, and the lattice connected with~$O_7(3)$},
     journal = {Sbornik. Mathematics},
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     volume = {74},
     number = {1},
     year = {1993},
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     url = {http://geodesic.mathdoc.fr/item/SM_1993_74_1_a11/}
}
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V. P. Burichenko. On a~special loop, the discon form, and the lattice connected with~$O_7(3)$. Sbornik. Mathematics, Tome 74 (1993) no. 1, pp. 145-167. http://geodesic.mathdoc.fr/item/SM_1993_74_1_a11/