Dear Frank J Hall

Basically, It is shown that every full 4 × 4 sign pattern allows J-orthogonality and as a result that, for n ≤ 4, all n × n full sign patterns allow a J-orthogonal matrix as well as a G-matrix.

Kind Regards

Qamar Ul Islam

Of course this condition is a base for differentiable manifolds, with many application or modelling based framework for analytic manifolds, such as Grassmann as well as Stiefel manifolds. Indeed, a useful work on the related topic is here : Chapter Sign pattern matrices