Siegel transformations for even characteristic. (Q1765911)

Let \(\text{SO}(V)\) be a special orthogonal group over a field \(K\) of characteristic 2. For \(\pi\in\text{SO}(V)\), \(B(\pi)\) denotes the image \(V(\pi-1)\). Under some conditions on \(K\) and the quadratic form, it is shown that every \(\pi\in\text{SO}(V)\) with \(\dim(B(\pi))=2d\) has the form \(\pi=\rho_1\cdots\rho_{d-1}\kappa\) with \(\rho_i\) being Siegel transformations and \(\dim(B(\kappa))=2\). An answer is given to the question whether \(\kappa\) above can be restricted to be a Siegel transformation.