On bireflectionality and trireflectionality of orthogonal groups (Q1091472)

Let V be a vector space endowed with a quadratic form Q. The \(f(x,y)=Q(x+y)-Q(x)-Q(y)\) is a bilinear form. The vector x is perpendicular to the vector y if \(f(x,y)=0\). The radical of V consists of all vectors that are perpendicular to all vectors in V. The weak orthogonal group \(O^*(V)\) consists of all form preserving bijections that keep every vector in the radical fixed. The author assumes that the field of scalars has characteristic distinct from 2. He shows (A) Every element in \(O^*(V)\) is a product of at most three involutions and (B) Every element in \(O^*(V)\) is a product of two involutions if and only if the index of V or the dimension of the radical is at most 1. This corrects an error in \textit{E. W. Ellers}, \textit{R. Frank}, and \textit{W. Nolte} [J. Algebra 88, 63-67 (1984; Zbl 0533.20020)].