How to distinguish between the latently real matrices and the block quaternions?

Let a complex n × n matrix A be unitarily similar to its entrywise conjugate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \bar{A} $\end{document}. If the unitary matrix P in the relation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \bar{A} = {P^*}AP $\end{document} can be chosen symmetric (skew-symmetric), then A is called a latently real matrix (respectively, a generalized block quaternion). Only these two cases are possible if A is a (unitary) irreducible matrix. The following question is discussed: How to find out whether a given A is a latently real matrix or a generalized block. quaternion? Bibliography: 5 titles.