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 II 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(alpha )= g* = (g(t))(-1) on E, algebra structure on A and a quadratic map (Q) over cap : A -> A. Then we prove the following results: (a) (Q) over cap (x(g)) = Q(x)(g)* for all x is an element of A and g is an element of E. (b) For x, y, z is an element of A and g is an element of 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 isometrics of T coincides with the group G* = (g is an element of GL(A) vertical bar a(g)b(g) = (ab)(g)* }. (2) The group G(0) = {g is an element of GL(A) vertical bar (Q) over cap (a(g)) = (Q) over cap (a)(g)* } is intermediate between E and G. (3) The group E = E* = {g* = (gt)(-1) vertical bar g is an element of E} .