Abstract
Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group ๐ช(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is ๐ช(V, I) = ๐ช(n). There is a nice result that ๐ช(n) โ Aut(๐ฌ(n)) over โ or โ, where ๐ฌ(n) is the Lie algebra of n ร n alternating matrices over the field. How about another field The answer is โYesโ if it is GF(2). We show it explicitly with the combinatorial basis โญ. This is a verification of Steinberg's main result in 1961, that is, Aut(๐ฌ(n)) is simple over the square field, with a nonsimple exception Aut(๐ฌ(5)) โ ๐ช(5) โ ๐6.
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Communicated by I. Shestakov.