Table[
Print[i,j," ",ReducePrimitives[
Der[Y[1,2,2,H[1]]]@Ar[i,j] + Der[Ar[i,j]]@Y[1,2,2,H[1]]
]],{i,3},{j,3}
]

{i,j}={2,1}

Der[Y[1,2,2,1]]@Ar[i,j]

Der[Y[1,2,2,1]]

Table[
Print[i,j,ReducePrimitives[
Der[Y[1,2,3,H[1]]]@Ar[i,j] + Der[Ar[i,j]]@Y[1,2,3,H[1]]
]],{i,4},{j,4}
]

{i,j}={1,3}

Der[Y[1,2,3,H[1]]]@Ar[i,j]

Der[Ar[i,j]]@Y[1,2,3,H[1]]

Der[Y[1,2,3,H[1]]]

Der[Y[1,2,3,H[1]]]@Ar[1,3]

CanonicalForm[S[sigma[1,2]]]

S[Exp[Ar[1,3]+Ar[2,3]+Y[1,2,3,H[bch]]]]

S[sigma[1,3],sigma[2,3]]

Factor[(((x[1])/((((-1+(E^(x[1]+x[2]))))^2) (x[1]+x[2])))-((2 (E^(x[1]+x[2])) x[1])/((((-1+(E^(x[1]+x[2]))))^2) (x[1]+x[2])))+(((E^(2 x[1]+2 x[2])) x[1])/((((-1+(E^(x[1]+x[2]))))^2) (x[1]+x[2])))+((x[2])/((((-1+(E^(x[1]+x[2]))))^2) (x[1]+x[2])))-((2 (E^(x[1]+x[2])) x[2])/((((-1+(E^(x[1]+x[2]))))^2) (x[1]+x[2])))+(((E^(2 x[1]+2 x[2])) x[2])/((((-1+(E^(x[1]+x[2]))))^2) (x[1]+x[2]))))]