This matrix should extend the control over the degrees of freedom of orthogonal matrices and offer a fresh perspective for applications in polarization modulation.

J-orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products, including, in particular, the downdating of Cholesky factorizations.

A. T. James, Normal Multivariate Analysis and the Orthogonal Group, The Annals of Mathematical Statistics, Vol. 25, No. 1 (Mar., 1954), pp. 40-75 ...