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}

{x1,x2,x3}={a[1,1,2],a[1,2,3],a[h[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])],1,2]}

B[x1,x2]+B[x2,x1]

B[x1,B[x2,x3]] + B[x2,B[x3,x1]] + B[x3,B[x1,x2]] // LSimp

B[x1,B[x2,x3]] + B[x2,B[x3,x1]] + B[x3,B[x1,x2]] // ColumnForm

B[x2,x3]

B[x1,γ[h[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])],1,2,3]]

B[x3,x1]

B[x1,x2]

B[x1,B[x2,x3]]

{y1,y2,y3}={a[1,1,2],a[1,2,3],a[(Subscript[b, 3]),1,2]};
B[y1,B[y2,y3]] + B[y2,B[y3,y1]] + B[y3,B[y1,y2]]

B[y2,y3]

B[y1,B[y2,y3]]