(* delta and gamma *)
δ /: δ[{i_,j_}]f_[{x___,j_,y___}]:=f[{x,i,y}];
δ /: δ[{i_,j_}]f_[{x___,i_,y___}]:=f[{x,j,y}];
δ /: δ[{i_,j_}]f_[{v___,{x___,j_,y___},w___}]:=f[{v,{x,i,y},w}];
δ /: δ[{i_,j_}]f_[{v___,{x___,i_,y___},w___}]:=f[{v,{x,j,y},w}];
δ /: δ[{i_,i_}]:=11;(* dimension dependent *)
δ /: δ[{i_,j_}]:=δ[{j,i}] /; {i,j}=!=Sort[{i,j}];
γ /: γ[list_]:=Signature[list]γ[Sort[list]] /; list=!=Sort[list];
γ /: γ[{}]:=1;
γ /: γ[{___,z_,___,z_,___}]:=0;
γ /: γ[{x1_,x2_,x3_,x4_,x5_,x6_,x7_,x8_,x9_,x10_,x11_,x12_,___}]:=0;(* dimension dependent *)
DF /: DF[{{x1_},{___,z_,___,z_,___}}]:=0;
F4 /: F4[{___,z_,___,z_,___}]:=0;
F4 /: F4[list_]:=Signature[list]F4[Sort[list]] /; list=!=Sort[list];

δ[{a,b}]F[{{p,q},{a,c}}]δ[{d,c}]

γ[{j,i,k,l}]

F4[{a,a,b,c}]

(* product of γ[{(Subscript[a, 1])...(Subscript[a, m])}] and γ[{(Subscript[b, 1])...(Subscript[b, n])}]: (γ^((Subscript[a, 1])...(Subscript[a, m])))(Subscript[γ, (Subscript[b, 1])...(Subscript[b, n])])=(∑_(k=m+n-11)^(Min(m,n)))(((-1))^(m k-((k(k+1))/2)))((m!n!k!)/((m-k)!k!(n-k)!k!))(Subscript[δ, [(Subscript[b, 1])...(Subscript[b, k])]^([(Subscript[a, 1])...(Subscript[a, k])))(Subscript[(γ^((Subscript[a, k+1])...(Subscript[a, m])])), (Subscript[b, k+1])...(Subscript[b, n])]]) *)
GammaGamma /: GammaGamma[gind1_,gind2_]:=
Module[{len1,len2,max,min,sgn0,id,gml,akl,a1,a2,bkl,b1,b2,b1pm,sgn1},

(* gind1={(Subscript[a, 1])...(Subscript[a, m])}, gind2={(Subscript[b, 1])...(Subscript[b, n])} *)
len1=Length[gind1];
len2=Length[gind2];
max=Min[len1,len2];
min=Max[1,len1+len2-11];(* dimension dependent *)
sgn0=Signature[gind1]Signature[gind2];

id=0;
gml[++id]=γ[Join[gind1,gind2]];(* gml[1]=(Subscript[(γ^((Subscript[a, 1])...(Subscript[a, m]))), (Subscript[b, 1])...(Subscript[b, n])]), k=0 *)
Do[akl=Subsets[gind1,{k99}];(* akl={{(Subscript[a, 1]),...,(Subscript[a, k])},{(Subscript[a, 1]),...,(Subscript[a, k-1]),(Subscript[a, k+1])},...}, Length[akl]=(Subscript[, m])(Subscript[C, k]) *)
     bkl=Subsets[gind2,{k99}];(* bkl={{(Subscript[b, 1]),...,(Subscript[b, k])},...},{(Subscript[b, 1]),...,(Subscript[b, k-1]),(Subscript[b, k+1])}, Length[bkl]=(Subscript[, n])(Subscript[C, k]) *)
     Do[a1=akl[[i99]];a2=Complement[gind1,akl[[i99]]];(* a1={(Subscript[a, 1]),...,(Subscript[a, k])}, a2={(Subscript[a, k+1]),...,(Subscript[a, m])} *)
          b1=bkl[[j99]];b2=Complement[gind2,bkl[[j99]]];(* b1={(Subscript[b, 1]),...,(Subscript[b, k])}, b2={(Subscript[b, k+1]),...,(Subscript[b, n])} *)
          b1pm=Permutations[b1];(* b1pm={{(Subscript[b, 1]),...,(Subscript[b, k])},{(Subscript[b, 1]),...,(Subscript[b, k]),(Subscript[b, k-1])},...} *)
          sgn1=Signature[Join[a1,a2]];
          gml[++id]=(((-1))^((k99(2len1-k99-1))/2))sgn0 sgn1(∑_(m99=1)^(k99!))(Signature[Join[b1pm[[m99]],b2]](∏_(l99=1)^k99)δ[{a1[[l99]],b1pm[[m99,l99]]}])γ[Join[a2,b2]];(* signature is assigned compared to the orginal position *)
   ,{i99,1,Length[akl]},{j99,1,Length[bkl]}];
,{k99,min,max}];

Expand[Sum[gml[i99],{i99,1,id}]]
];

GammaGamma[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11}]

GammaGamma[{b},{a}]+GammaGamma[{a},{b}]

GammaGamma[{a,b},{d,c}]

GammaGamma[{a,b},{b}]

GammaGamma[{c},{b,a}]

GammaGamma[{a,b},{c,d,e}]

(* GammaProd: Product of (Subscript[A, i])(Subscript[γ, i]) and (Subscript[B, j])(Subscript[γ, j]) *)

GammaProd /: GammaProd[T1_,T2_]:=
Module[{const1,const2,res,T1mC,T2mC,len1,len2,Cf1,Cf2,T1Gm,T2Gm},

(* T1=const1+(Subscript[A, i])(Subscript[γ, i]), T2=const2+(Subscript[B, j])(Subscript[γ, j]) *)
const1=T1 /. γ[___]->0;
const2=T2 /. γ[___]->0;

(* case: (T1==const1)||(T2==const2) *)
If[(T1==const1)||(T2==const2),res=Expand[T1*T2];];

(* case: (T1=!=const1)&&(T2=!=const2) *)
If[(T1=!=const1)&&(T2=!=const2),
    T1mC=Expand[T1-const1];(* T1mC=(Subscript[A, i])(Subscript[γ, i]) *)
    len1=Length[Position[T1mC,γ]];(* len1=number of γ in T1 or T1mC *)
    If[len1==1,
         Cf1={T1mC /. γ[___]->1};(* Cf1={(Subscript[A, 1])} *)
         T1Gm={T1mC/Cf1[[1]]};(* T1Gm={(Subscript[γ, 1])} *),
         Cf1=Table[T1mC[[i99]] /. γ[___]->1,{i99,1,len1}];(* Cf1={(Subscript[A, 1]),...,(Subscript[A, m])} *)
         T1Gm=Table[T1mC[[i99]]/Cf1[[i99]],{i99,1,len1}];(* T1Gm={(Subscript[γ, 1]),...,(Subscript[γ, m])} *)
  ];
    T2mC=Expand[T2-const2];(* T2mC=(Subscript[B, i])(Subscript[γ, i]) *)
    len2=Length[Position[T2mC,γ]];(* len2=number of γ in T2 or T2mC *)
    If[len2==1,
         Cf2={T2mC /. γ[___]->1};(* Cf2={(Subscript[B, 1])} *)
         T2Gm={T2mC/Cf2[[1]]};(* T2Gm={(Subscript[γ, 1])} *),
         Cf2=Table[T2mC[[j99]] /. γ[___]->1,{j99,1,len2}];(* Cf2={(Subscript[B, 1]),...,(Subscript[B, n])} *)
         T2Gm=Table[T2mC[[j99]]/Cf2[[j99]],{j99,1,len2}];(* T2Gm={(Subscript[γ, 1]),...,(Subscript[γ, n])} *)
  ];
res=Expand[const1*const2+const1*T2mC+const2*T1mC+Sum[Cf1[[i99]]Cf2[[j99]]GammaGamma[T1Gm[[i99,1]],T2Gm[[j99,1]]],{i99,1,len1},{j99,1,len2}]];
];

Expand[res]
];

GammaProd[2γ[{b}]+3γ[{a}],4γ[{c}]]

GammaProd[1+2γ[{b}]+3γ[{a}],10+4γ[{c}]]

GammaGamma[{a,b,c},{f}]

GammaGamma[{a,b,c},{f}] /. f->a

GammaProd[γ[{a,b,c}]+γ[{b}]+3γ[{a}],4γ[{c}]+3γ[{c,d}]]

GammaProd[10+γ[{a,b,c}],12]

GammaProd[21,10+γ[{c,b}]]

Expand[GammaProd[-(1/36)δ[{μ,i1}]γ[{j1}]+(1/288)γ[{μ,i1,j1}],-(1/36)δ[{ν,i2}]γ[{j2}]+(1/288)γ[{ν,i2,j2}]]]

GammaProd[γ[{μ,i1,j1,k1,l1}],γ[{ν,i2,j2,k2,l2}]]

Expand[GammaProd[-(1/36)F[{a,i,j,k}]γ[{i,j,k}]-(1/288)F[{i,j,k,l}]γ[{a,i,j,k,l}],γ[{b,c}]]]

Expand[GammaProd[-(1/36)DF[{a,b,i,j,k}]γ[{i,j,k}]-(1/288)DF[{a,i,j,k,l}]γ[{b,i,j,k,l}],γ[{m}]]]

(* Use antisymmetric property *)

Antisym /: Antisym[T1_,asm0_]:=
Module[{fn,tmall,sb2,cf,list,rep,nb1,nb2,asm,sgn,tm},

fn={p1,p2,p3,p4,p5,p6,p7,p8,p9};(* contracted indices *)
sb2=Subsets[asm0,{2}];(* asm0={a,b,c}; sb2={{a,b},{a,c},{b,c}} *)
tmall=1+T1 /. Table[δ[sb2[[u]]]->0,{u,1,Length[sb2]}];(* remove δ[{a,b}] etc *)

Do[cf=tmall[[u]] /. {f_[x_]->1};
     list=Flatten[tmall[[u]]/cf /. {Times->List,f_[x_]->x}];
     rep=Normal[Counts[list]];
     nb1=0;nb2=0;
     asm={};(* antisym indices as appeared *)
     Do[Which[MemberQ[asm0,rep[[v,1]]],rep[[v,2]]=asm0[[++nb1]];asm=Join[asm,{rep[[v,1]]}];,(* align antisym indices *)
                   FreeQ[asm0,rep[[v,1]]]&&rep[[v,2]]==2,rep[[v,2]]=fn[[++nb2]];,(* align paired indices *)
                   FreeQ[asm0,rep[[v,1]]]&&rep[[v,2]]==1,rep[[v]]=Hold[Nothing];(* remove non-antisym single index from rep *)
     ];
   ,{v,1,Length[rep]}];
     rep=ReleaseHold[rep];
     sgn=Signature[asm]Signature[asm0];
     tm[u]=sgn*tmall[[u]] /. rep;
,{u,2,Length[tmall]}];

Sum[tm[u],{u,2,Length[tmall]}]
];

Antisym /: Antisym[T1_,asm0_,asm1_]:=
Module[{tmall},

tmall=Antisym[T1,asm0];
tmall=Antisym[tmall,asm1];

tmall
];

tmp=Expand[F4[{c,k,j,i}]GammaGamma[{b,a},{k,j,i}]]

Antisym[tmp,{a,b,c}]

tmp=Expand[-(1/36)F4[{a,i,j,k}](GammaGamma[{b,c},{i,j,k}]+GammaGamma[{i,j,k},{b,c}])+(1/288)F4[{i,j,k,l}](GammaGamma[{b,c},{a,i,j,k,l}]-GammaGamma[{a,i,j,k,l},{b,c}])]

Antisym[tmp,{a,b,c}]

Antisym[GammaGamma[{i,j,k},{b,c}],{i,j,k},{b,c}]

Antisym[GammaGamma[{a1,a2,a3,a4,a5,a6},{b1,b2,b3}],{a1,a2,a3,a4,a5,a6},{b1,b2,b3}]