Z1=Z1.nb
Check that the two expressions for Z are equivalent.

(* the following function changes sign of kappa[a,b,i,j] so that a<b and i<j *) kappaGeneral [a_,b_,i_,j_]:= Signature[{a,b}] Signature[{i,j}]κ[Min[{a,b}],Max[{a,b}],Min[{i,j}],Max[{i,j}]]

Constraints for 

  	Z1 = {k in A^2_2  :  k(u) /\ v = u /\ k(v) for all u,v and
  			          trace(k)=0 }

ConstraintsZ1={};

(* first add conditions for ku /\ v = u /\ kv *)
For[i=1, i≤4,i++,
For[j=1, j≤4,j++,
For[a=1, a≤4,a++,
For[b=1, b≤4,b++,
Exp1=Sum[Signature[{i,j,p,q}]kappaGeneral[p,q,a,b],{p,1,4},{q,1,4}];
Exp2=Sum[ Signature[{a,b,p,q}] kappaGeneral[p,q,i,j] ,{p,1,4},{q,1,4}];
If[Simplify[Exp1==Exp2, ConstraintsZ1],(*true*), (*false*), (*otherwise*)
ConstraintsZ1 = Append[ConstraintsZ1, Exp1==Exp2]
];
]
]
]
]

(* check that condition trace k=0 is independent of these conditions. If we obtain True, then it is a consequence of the above conditions. *)
Simplify[0== Sum[ kappaGeneral[i,j,i,j] ,{i,1,4},{j,1,4}], ConstraintsZ1]

(* add trace constraint, as we did not get True. *)
ConstraintsZ1 = Append[ConstraintsZ1, 
0== Sum[ kappaGeneral[i,j,i,j] ,{i,1,4},{j,1,4}]
];

(* List all constraints *)
ConstraintsZ1 //MatrixForm

Length[ConstraintsZ1]

Constraints for 

  	Z2 = {k in A^2_2  :  k^ij_lj = 0 }

ConstraintsZ2={};

For[i=1, i≤4,i++,
For[j=1, j≤4,j++,
ConstraintsZ2 = Append[ConstraintsZ2, 
0== Sum[ kappaGeneral[i,P,j,P] ,{P,1,4}]
]
]
]
ConstraintsZ2 //MatrixForm

Length[ConstraintsZ2]

Show that all constraints in Z1 are satisfied if constraints in Z2 hold, that is, 

	Z2 subset Z1

out={};
For[temp=1, temp≤Length[ConstraintsZ1], temp++,
out=Append[out, 
Simplify[ConstraintsZ1[[temp]], ConstraintsZ2]];
]
out

Show that all constraints in Z2 are satisfied if constraints in Z1 hold, that is, 

	Z1 subset Z2

out={};
For[temp=1, temp≤Length[ConstraintsZ2], temp++,
out=Append[out, 
Simplify[ConstraintsZ2[[temp]], ConstraintsZ1]];
]
out

We have shown that Z1=Z2