From what I understand, it follows from the ugly lemma (page 135 Cassels and Frohlich) that $H^2(Gal(K^{un}/K), {K^{un}}^{\times}) \cong H^2(Gal(K^{sep}/K), {K^{sep}}^\times) $. One can then define invariant map for $H^2(Gal(K^{sep}/K), {K^{sep}}^\times) $ and get the second inequality of the local class field theory.
If this is so, why does Serre (in his article in Cassels and Frohlich page 134) constructs open subgroups of units with trivial cohomology (I think Neukirch calls it class field axiom ) to prove the second inequality?
Am I missing something here? I know the ugly lemma is useful in Global class field theory as well.