Pensieve Header: Verifying the B4 Syzygy.

Unprotect[NonCommutativeMultiply];
0**_ = 0;
_**0 = 0;
(c_?NumberQ * a_) ** b_ := Expand[c*(a**b)];
a_ ** (c_?NumberQ * b_) := Expand[c*(a**b)];
a_Plus ** b_ := (# ** b)& /@ a;
a_ ** b_Plus := (a ** #)& /@ b;
b[x_,y_] := x**y-y**x;
loc={
η[i_,j_,k_] :> b[a[i,j], a[i,k]]+b[a[i,j],a[j,k]] + b[a[i,k],a[j,k]],
γ[i_,j_,k_,l_] :> b[a[i,j],a[k,l]]
};
glob={
η[i_,j_,k_] :> Plus[
b[a[i,j], a[i,k]]+b[a[i,j],a[j,k]] + b[a[i,k],a[j,k]],
a[i,j]**a[i,k]**a[j,k] - a[j,k]**a[i,k]**a[i,j]
],
γ[i_,j_,k_,l_] :> b[a[i,j],a[k,l]]
};
syz=Plus[
+b[η[2,3,4], +a[1,2]+a[1,3]+a[1,4]],
-b[η[1,3,4], -a[1,2]+a[2,3]+a[2,4]],
+b[η[1,2,4], -a[1,3]-a[2,3]+a[3,4]],
-b[η[1,2,3], -a[1,4]-a[2,4]-a[3,4]],
-b[γ[1,2,3,4], a[1,3]+a[1,4]+a[2,3]+a[2,4]],
+b[γ[1,3,2,4], a[1,2]+a[1,4]-a[2,3]+a[3,4]],
-b[γ[1,4,2,3], a[1,2]+a[1,3]-a[2,4]-a[3,4]]
];

syz /. {η[i_,j_,k_] :> (Subscript[η, ({{i, j, k}})]), γ[i_,j_,k_,l_] :> (Subscript[γ, ({{k, l}})]^({{i, j}}))}; (* failed *)

syz /. loc

syz /. glob