Summary: Starting with a \(p\)-local space \(X\) of \(l\) odd dimensional cells, \(l<p-1\), G. Cooke, J. R. Harper, and A. Zabrodsky [Topology 18, 349–359 (1979; Zbl 0426.55009)] constructed an \(H\)-space \(Y\) with the property that \(H_*(Y)\) is generated as an exterior Hopf algebra by \(\widetilde H^*(X)\). F. Cohen and J. Neisendorfer [Algebraic topology, Proc. Conf., Aarhus 1982, Lect. Notes Math. 1051, 351–359 (1984; Zbl 0582.55010)], and later P. Selick and J. Wu [Mem. Am. Math. Soc. 701, 109 p. (2000; Zbl 0964.55012)], reproduced this result with different constructions. We use the Selick and Wu approach to show that \(Y\) is homotopy associative and homotopy commutative if \(X\) is a suspension and \(l<p-2\).