Generalized Hurwitz maps of the type \(S\times V\to W\) (Q1373530)

Let \(S, V\), and \(W\) be real vector spaces with nonsingular real scalar products which are pseudo-Euclidean or symplectic. A \(J^3\)-mapping is a bilinear mapping \(S\times V\to W:(s,v)\to s\cdot v\) such that \((a,b)_S(x,y)_V={1\over2}[(a\cdot x,b\cdot y)_W+\varepsilon(b\cdot x,a\cdot y)_W]\) for \(x,y\in V\) and \(a,b\in S\), \(\varepsilon=1\) or \(-1\). This is a generalized Hurwitz condition. Under certain assumptions the authors give a number of characterizations of \(J^3\)-mappings, they derive relations between the modified structure matrices of a \(J^3\)-triple, get generators of a real Clifford algebra related to a certain metric tensor, and obtain information on generalized Hurwitz pairs.