Minkowski half-turns (Q583609)

Let V be a finite-dimensional orthogonal vector space over a commutative field with characteristic \(\neq 2\). Assume that the radical is 1- dimensional. It was shown by \textit{E. Ellers} [Geom. Dedicata 15, 363-375 (1984; Zbl 0539.51007)] that the so called half-turns (a special type of involutions of O(V)) generate the subgroup of O(V) with determinant 1. The authors give a self-contained solution for the corresponding length- problem, i.e. the problem of finding the minimal number of half turns needed for generating a given transformation.