ORTHOGONAL GROUPS O(N) OVER GF(2) AS AUTOMORPHISMS

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group ?(V, Q) is defined as automorphisms on (V, Q). If Q=I, it is ?(V, I)=?(n). There is a nice result that ?(n)Aut(?(n)) over or C, where ?(n) is the Lie algebra of nxn alternating matrices over the field. How about another field The answer is Yes if it is GF(2). We show it explicitly with the combinatorial basis C. This is a verification of Steinberg's main result in 1961, that is, Aut(?(n)) is simple over the square field, with a nonsimple exception Aut(?(5))?(5)?(6).