Computation for Proposition 3.1 ii) => i)

SetDirectory["~/writing/WIP/KappaLib/"];
<<KappaLib.m

kappa = emGeneralKappa["kappa"];
emKappaToMatrix[kappa]//MatrixForm

rules=Table[
Sum[emReadNormal[kappa,c,r,a,b] Signature[{d,s,a,b}],
{a,1,4},{b,1,4}]
+Sum[emReadNormal[kappa,d,r,a,b] Signature[{c,s,a,b}],
{a,1,4},{b,1,4}]==0
,
{c,1,4},{d,1,4},{r,1,4},{s,1,4}];

rules = Union[Flatten[rules]]

(* 

  Simplify constraints:
 
    Simplify[..]  - removes leading constants (like -2 kappa31=0)
  Union[..]  - remove duplicate entries
*)
assumptions=Union[Flatten[Simplify[rules]]];
assumptions

kappaTrace=emTrace[kappa];
Simplify[kappa==kappaTrace/6 emIdentityKappa[],assumptions]