Siegel transformations for even characteristic

Abstract

Let V be a vector space over a field K of even characteristic and ∣ K∣ > 3. Suppose K is perfect and π is an element in the special orthogonal group SO( V) with dim B( π)=2 d. Then π = ρ 1 ⋯ ρ d−1 κ, where ρ j , j = 1 ,…, d − 1, are Siegel transformations and κ ∈ SO( V) with dim B( κ) = 2. The length of π with respect to the Siegel transformations is d if π is unipotent or if dim B ( π)/rad B( π) ⩾ 4; otherwise it is d + 1.