Suggested Standard Forms for Certain Real Hadamard Matrices

Any Hadamard matrix constructed by methods of R. E. A. C. Paley can be converted easily by one or two elementary matrix operations to a form sharing at least four desirable properties with standard forms of Walsh matrices: symmetry, zero trace, normal form, and the same number of 1´s as −1´s in every row and column except the 0th row and 0th column.