ON A SPECIAL LOOP, THE DICKSON FORM, AND THE LATTICE CONNECTED WITH O7(3)

A special commutative Moufang loop 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 E6. Next, with the help of H, a 27-dimensional representation is constructed for O7(3) over Z[zeta] , zeta3 = 1 . This makes it possible to prove anew the embedding O7 (3) subset-or-equal-to 2 E6(2). A similar construction concerning the embedding L4(3) subset-or-equal-to F4(2) is described.