Explicit constructions of loops with commuting inner mappings

In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. As of now, no other examples are known. We construct many such loops from groups of nilpotency class two by replacing the product x y with x y h in certain positions, where h is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.