Show that KappaII in perp(U)

KappaNorm[a_,b_,i_,j_]:= Signature[{a,b}] Signature[{i,j}] κ[Min[{a,b}],Max[{a,b}],Min[{i,j}],Max[{i,j}]]

KappaDash[i_,j_]:= Sum[KappaNorm[i,mmm,j,mmm],{mmm,1,4}] - 1/4 Sum[KappaNorm[ii,jj,ii,jj],{ii,1,4},{jj,1,4}] KroneckerDelta[i,j]

tt[i_,k_,j_,l_]:= KappaDash[i,j] KroneckerDelta[k,l]

KappaII[i_,j_,l_,m_]:= 1/2 ( tt[i,j,l,m]-tt[i,j,m,l]-tt[j,i,l,m]+tt[j,i,m,l])

Sum[ KappaII[i,j,l,m] 1/2(KroneckerDelta[l,i] KroneckerDelta[m,j]-KroneckerDelta[l,i] KroneckerDelta[m,j]),{i,1,4},{j,1,4},{l,1,4},{m,1,4}]