A proof of the Adem relations

We give an alternative proof of the Bullett-Macdonald identity for the Steenrod squares, which is in turn equivalent to the Adem relations. The main idea is to show that the iterated total squaring operation S2 : Hn(X) → H4n(X × BZ2 × BZ2) is the restriction of a total fourth-power operation T : Hn(X) → H4n(X × BΣ4).