Computations related proof of Theorem 4.3

Claim 1: If F^(ij) xi_ij = 0, then xi is of the form

         (({{0, AA, BB, CC}, {-AA, 0, DD, -BB}, {-BB, -DD, 0, EE}, {-CC, BB, -EE, 0}}))

(* In general xi is of the form *)
XI={{0,AA,BB,CC},
{-AA,0,DD,-GG},
{-BB,-DD,0,EE},
{-CC,GG,-EE,0}};
XI//MatrixForm
XI+Transpose[XI]

Fmat={{0,0,1,0},
{0,0,0,1},
{-1,0,0,0},
{0,-1,0,0}};
Fmat//MatrixForm

Simplify[Sum[Fmat[[i]][[j]] XI[[i]][[j]],{i,1,4},{j,1,4}]==0]

(* Thus: if F^(ij) xi_ij=0, then BB=GG and the claim follows *)

Claim 2: If xi is as above and

       ξ_pi F^ij G_jq = ξ_qi F^ij G_jp
  
  then xi=0

(* By the above argument we assume that xi is of the form *)
XI={{0,AA,BB,CC},
{-AA,0,DD,-BB},
{-BB,-DD,0,EE},
{-CC,BB,-EE,0}};
XI//MatrixForm
XI+Transpose[XI]

Gmat={{0,G12,G13,G14},
{-G12,0,G23,G24},
{-G13,-G23,0,G34},
{-G14,-G24,-G34,0}};
Gmat//MatrixForm
Gmat+Transpose[Gmat]

(* define inverse of F *)
Finv= Inverse[Fmat];
Finv//MatrixForm

XiConstraints =Table[
Sum[
XI[[p]][[i]] Finv[[i]][[j]] Gmat[[j]][[q]]-XI[[q]][[i]] Finv[[i]][[j]] Gmat[[j]][[p]],
{i,1,4},{j,1,4}
]==0,
{p,1,4},{q,1,4}
];
XiTable=Flatten[FullSimplify[XiConstraints]];
XiTable//MatrixForm

(* remove redundant equations *)
NewConstraints = {};
For[i=1, i≤ Length[XiTable], i++,
OneConstr = XiTable[[i]];
If[Simplify[OneConstr, NewConstraints]==True, 
(* true, do nothing *)
Print[ToString[i] <>": Removing redundant constraint " <> ToString[OneConstr]],
(* false (?)*)
 Print["??"],
(* not true/false, that is, a new constraint *)
NewConstraints = Append[NewConstraints, OneConstr];
Print[ToString[i] <>": Adding new constraint " <> ToString[OneConstr]]

]
]

NewConstraints//MatrixForm

Claim 3: Writing out F v G neq 0 gives G13 + G24 neq 0:

Simplify[Sum[Signature[{i,j,k,l}]Gmat[[i]][[j]] Fmat[[k]][[l]], {i,1,4},{j,1,4},{k,1,4},{l,1,4}]==0]