Quadratic Forms on the 27-Dimensional Modules for E6 in Characteristic Two

The purpose of this paper is to study the Chevalley group E of type E 6 (K) over fields K of characteristic two.We use the generalized quadrangle (P, L) over K of type O - 6 (2) to construct a trilinear form T on a 27-dimensional vector space A, this form preserves the action of E .We introduce an involution g g = g * = (g t ) -1 on E , algebra structure on A and a quadratic map Q : A A. Then we prove the following results: (a) Q(x g ) = Q(x) g * for all x A and g E .(b) For x, y, z A and g E , the following holds true:(1) x g y g = (xy) g * , and ( 2) T (x g , y g , z g ) = T (x, y, z) .(c) The main results:(1) The group G of isometries of T coincides with the group(3) The group E = E * = {g * = (g t ) -1 | g E} .