Orthogonal Cayley-Klein groups
The author calls the following setting a Cayley-Klein vector space. In a finite-dimensional vector space \(V\neq\{0\}\) over a field of characteristic \(\neq 2\) let \(V= R_0\supset R_1\supset\cdots\supset R_r= \{0\}\) be a proper chain of subspaces and \(f_1,\dots, f_r\) symmetric bilinear forms such that \(f_i: R_{i-1}\times R_{i-1}\to K\) and \(\text{rad\,}f_i= R_i\) for \(i= 1,\dots,r\). Vectors \(a,b\in V\) are called orthogonal if \(a,b\in R_{i-1}\) implies that \(f_i(a, b)= 0\) for \(i= 1,\dots, r\). In the group of automorphisms of a Cayley-Klein vector space each automorphism is a product of involutions with one-dimensional negative spaces. This corresponds to the fact that each element of an orthogonal group is a product of symmetries.
scientific article; zbMATH DE number 5009763
scientific article