Construction of symmetric and symplectic bilinear forms (Q1303806)

Let \(V\) be a finite-dimensional vector space over a field \(K\), \(\text{char} K\neq 2\). Let \(S\) be a set of simple involutory mappings in \(GL(V)\). Assume \(S\) is invariant under conjugation by elements in \(S;\) if \(\sigma,\tau\in S\) and \(B(\sigma)=B(\tau)\), then \(\sigma=\tau;\) if \(\sigma_1,\sigma_2,\sigma_3\in S\), then the negative space of \(\sigma_1\sigma_2\sigma_3\) is distinct from zero; there are elements \(\sigma_1,\ldots,\sigma_n,\rho\) in \(S\) such that \(B(\sigma_1)+\cdots+B(\sigma_n)=V\) and \(\rho\) does not commute with any \(\sigma_i\). The author shows that there is a symmetric bilinear form \(f\) on \(V\) such that the elements in \(S\) are reflections with respect to \(f\). Also, if \(S\) is a set of transvections in \(GL(V)\) satisfying conditions similar to those stated above, then there is a symplectic form such that the elements in \(S\) are symplectic transvections.