Looks like quasigroup elements have to be multiplicative elements (as that is not implied by ExtL and ExtRElement)

gap/quasigroups.gd

@@ -45,6 +45,7 @@ InstallTrueMethod(IsLeftQuotientElement, IsMultiplicativeElementWithInverse);
 
 ## element of a quasigroup
 DeclareCategory( "IsQuasigroupElement",
+                 IsMultiplicativeElement and 
                  IsLeftQuotientElement and IsRightQuotientElement );
 DeclareRepresentation( "IsQuasigroupElmRep",
     IsPositionalObjectRep and IsMultiplicativeElement, [1] );