Pensieve header: γ3-checks for OneCo. Branched from OneCoComputations.nb.

The bracket

In the basis {a[f,i,j], γ[f,i,j], γ[f,i,j,k], γa[f,i,j,k,l]}.

DistinctQ[is___] := (Sort[{is}]===Union[{is}]);

a[0,__] := 0;
a /: a[f_,is__] + a[g_, is__] := a[f+g, is];
a /: -a[f_, is__] := a[Expand[-f], is];
γ[0,__] :=0;
γ /: γ[f_,is__] + γ[g_, is__] := γ[f+g, is];
γ /: -γ[f_, is__] := γ[Expand[-f], is];
γa[0,__] :=0;
γa /: γa[f_,is__] + γa[g_, is__] := γa[f+g, is];
γa /: -γa[f_, is__] := γa[Expand[-f], is];

γ[f_,j_,k_,k_] = 0;
γ[f_,j_,k_,l_] /; OrderedQ[{l,k}]∧DistinctQ[k,l] := γ[-f, j, l, k];

B[0,_] =0; B[_, 0] = 0;
B[x_,x_] = 0;
B[x_Plus, y_] := B[#,y]& /@ x;
B[x_, y_Plus] := B[x,#]& /@ y;

[a,a] brackets:

B[a[f_, j_,k_], a[g_, l_,m_]] := Plus[
B[a[j,k], a[l,m]] /. {a[h_,is__] :> a[Expand[f g h], is], γ[h_,is__] :> γ[Expand[f g h], is]},
γa[Expand[f((∂_(Subscript[b, j]))g-(∂_(Subscript[b, k]))g)], j,k,l,m] + γa[Expand[g((∂_(Subscript[b, m]))f-(∂_(Subscript[b, l]))f)], j,k,l,m]
];
(* tt *) B[a[j_,k_], a[j_,l_]] /; DistinctQ[j,k,l] := γ[1,j,k,l];
(* hh *) B[a[j_,k_], a[i_,k_]] /; DistinctQ[i,j,k] := a[(Subscript[b, i]), j,k] - a[(Subscript[b, j]), i,k];
(* th *) B[a[j_,k_], a[i_,j_]] /; DistinctQ[i,j,k] := a[(Subscript[b, j]), i,k] - a[(Subscript[b, i]), j,k] + γ[1, i,j,k];
(* ht *) B[a[j_,k_], a[k_,l_]] /; DistinctQ[j,k,l] := -B[a[k,l], a[j,k]];
(* loc *) B[a[j_,k_], a[l_,m_]] /; DistinctQ[j,k,l,m] := 0;

[a,(Subscript[γ, 2])] brackets:

B[γ[f_, i_,j_], a[g_, k_,l_]] := -B[a[g, k,l], γ[f, i,j]];
(* ttγ *) B[a[f_, j_,k_], γ[g_, j_,l_]] /; DistinctQ[j,k,l] := 0;
(* hhγ *) B[a[f_, j_,k_], γ[g_, i_,k_]] /; DistinctQ[i,j,k] := - γ[(Subscript[b, j])f g, i,k];
(* thγ *) B[a[f_, j_,k_], γ[g_, i_,j_]] /; DistinctQ[i,j,k] := γ[(Subscript[b, j])f g, i,k];
(* htγ *) B[a[f_, j_,k_], γ[g_, k_,l_]] /; DistinctQ[j,k,l] := γ[(Subscript[b, j])f g, k,l] - γ[(Subscript[b, j])f g, j,l];
(* locγ *) B[a[f_, j_,k_], γ[g_, l_,m_]] /; DistinctQ[j,k,l,m] := 0;
B[a[f_, j_,k_], γ[g_, j_,k_]] :=  γ[-(Subscript[b, j])f g, j,k];

[a,(Subscript[γ, 3])] brackets:

(* ttγ3 *) B[a[f_, j_,k_], γ[g_, j_,l_,m_]] /; DistinctQ[j,k,l,m] := 0;
(* thγ3-1 *) B[a[f_, j_,k_], γ[g_, i_,j_,l_]] /; DistinctQ[i,j,k,l] := γ[(Subscript[b, j])f g,i,k,l] + γa[f g,i,l,j,k];
(* thγ3-2 *) B[a[f_, l_,k_], γ[g_, i_,j_,l_]] /; DistinctQ[i,j,k,l] := γ[-(Subscript[b, l])f g,i,k,j] + γa[-f g,i,j,l,k];
(* htγ3 *) B[a[f_, j_,k_], γ[g_, k_,l_,m_]] /; DistinctQ[j,k,l,m] := γ[-(Subscript[b, k])f g,j,l,m] + γ[(Subscript[b, j])f g,k,l,m];
(* hhγ3-1 *) B[a[f_, j_,k_], γ[g_, n_,i_,k_]] /; DistinctQ[n,i,j,k] := γ[-(Subscript[b, j])f g, n,i,k] + γa[f g, n,i,j,k];
(* hhγ3-2 *) B[a[f_, j_,i_], γ[g_, n_,i_,k_]] /; DistinctQ[n,i,j,k] := γ[(Subscript[b, j])f g, n,k,i] + γa[-f g, n,k,j,i];
(* a-th-1 *) B[a[f_, j_,k_], γ[g_, j_,k_,l_]] /; DistinctQ[j,k,l] := γa[-f g,j,k,j,l];
(* a-th-2 *) B[a[f_, j_,l_], γ[g_, j_,k_,l_]] /; DistinctQ[j,k,l] := γa[f g,j,l,j,k];
(* a-hh-1 *) B[a[f_, j_,k_], γ[g_, i_,j_,k_]] /; DistinctQ[i,j,k] := γ[-(Subscript[b, j])f g, i,j,k] + γa[f g, i,j,j,k] + γa[f g, i,k,j,k];
(* a-hh-2 *) B[a[f_, k_,j_], γ[g_, i_,j_,k_]] /; DistinctQ[i,j,k] := γ[(Subscript[b, k])f g, i,k,j] + γa[-f g, i,k,k,j] + γa[-f g, i,j,k,j];

[γ,γ] brackets:

B[_γ, _γ] := 0;

(Subscript[γ, 3]) checks

ca[0,__] :=0;
ca /: ca[f_,is__] + ca[g_, is__] := ca[f+g, is];
ca /: -ca[f_, is__] := ca[Expand[-f], is];
B[a[f_,j_,k_], c[l_]] /; DistinctQ[j,k,l]:= 0;
B[a[f_,j_,k_], c[j_]] := γ[-f,j,k];
B[a[f_,j_,k_], c[k_]] := γ[f,j,k];
B[x_, ca[f_, i_,j_,k_]] := Plus[
B[x, c[i]] /. γ[g_,l_,m_] :> γa[f g, l,m,j,k],
B[x, a[f,j,k]] /. {a[g_,l_,m_] :> ca[g, i,l,m], _γ -> 0, _γa -> 0}
];
δaa[f_, i_,j_,k_,l_] /; OrderedQ[{l,j}]∧DistinctQ[i,j,k,l] := δaa[f, k,l,i,j];
δaa[f_, i_,j_,i_,l_] /; OrderedQ[{l,j}]∧DistinctQ[j,l]:= δaa[f,i,l,i,j];
(* missing a 4T! *)
Eγ = {
γ[f_,j_,k_,l_] :> ca[f, l,j,k]+ca[Expand[-f], k,j,l],
γa[f_, i_,j_,k_,l_] :> δaa[f, i,j,k,l] + ca[Expand[-(Subscript[b, i])f], j,k,l]
};
Check3[y_γ] := {t1=B[a[f,j,k], y] /. Eγ, t2=B[a[f,j,k], y /. Eγ] /. Eγ, t1==t2 // Simplify}

tt check

γ[g,j,l,m] // Check3

th checks

γ[g, i,j,l] // Check3 // ColumnForm

γ[g, i,l,j] // Check3 // ColumnForm

ht check

γ[g, k,l,m] // Check3 // ColumnForm

hh checks

γ[g, n,i,k] // Check3 // ColumnForm

γ[g, n,k,i] // Check3 // ColumnForm

a-th checks

γ[g, j,l,k] // Check3 // ColumnForm

γ[g, j,l,k] // Check3 // ColumnForm

a-hh checks

γ[g, i,j,k] // Check3 // ColumnForm

γ[g, i,k,j] // Check3 // ColumnForm

x1 - x2 - x3 check

{x1,x2,x3}={a[f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])],1,2], a[g[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])],1,3],a[h[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])],2,3]} /. {_f|_g|_h :> 1}

B[x2,x3] /. Eγ

j1=B[x1,B[x2,x3] /. Eγ] /. Eγ /. _δaa :> 0

j2 = B[x2,B[x3,x1] /. Eγ]  /. Eγ/. _δaa -> 0

j3 = B[x3,B[x1,x2] /. Eγ]  /. Eγ/. _δaa -> 0

j1+j2+j3

B[x1,B[x2,x3]] /. Eγ/. _δaa :> 0

B[x2,B[x3,x1]]  /. Eγ/. _δaa -> 0

B[x3,B[x1,x2]]  /. Eγ/. _δaa -> 0

(B[x1,B[x2,x3]] + B[x2,B[x3,x1]] + B[x3,B[x1,x2]])

(B[x1,B[x2,x3]] + B[x2,B[x3,x1]] + B[x3,B[x1,x2]]) /. Eγ

γ[(Subscript[b, 1]),1,2,3]/. Eγ