An alternative approach to the Adem relations in the mod 2 Steenrod algebra

The Leibniz-Hopf algebra F is the free associative algebra over Z on one generator S-n in each degree n > 0, with coproduct given by Delta(S-n) = Sigma(i+j=n) S-i circle times S-j. We introduce a new perspective on the Adem relations in the mod 2 Steenrod algebra A(2) by studying the map pi* dual to the Hopf algebra epimorphism pi : F circle times Z/2 -> A(2). We also express Milnor's Hopf algebra conjugation formula in A(2)* in a different form and give a new approach for the conjugation invariant problem in A(2)*.