© 2017, University of Queensland. All rights reserved. A construction method for orthogonal ±1 matrices based on a variation of the Williamson array, first described by N. A. Balonin, on his web page mathscinet.ru/catalogue/propus/ where he called it the propus array, gives symmetric propus-Hadamard matrices using (Forumala presented). We show that: for q ≡ 1 (mod 4), a prime power, symmetric propus-Hadamard matrices exist for order 2(q + 1); and for q ≡ 1 (mod 4), a prime power, and ½(q + 1) a prime power or the order of the core of a symmetric conference matrix (this happens for q = 89), symmetric propus-type Hadamard matrices of order 4(2q + 1) exist. We give constructions to find symmetric propus-Hadamard matrices for 57 orders 4n, n < 200 odd.