VS=True;

Jacobi@@{a[f,1,1],a[g,1,2],a[h,2,1]}

Jacobi@@{a[f,1,1],a[g,1,2],a[h,2,1]} /. (ϵrules={(Subscript[ϵ, 5])->1,(Subscript[ϵ, 7])->-(Subscript[ϵ, 6])})

Jacobi@@{a[f,1,2],a[g,1,3],a[h,2,3]}

Jacobi@@{a[1,1,2],a[1,1,3],a[1,2,1]} /. ϵrules

Jacobi@@{a[1,1,2],a[1,1,3],a[1,2,1]} /. (ϵrules=ϵrules⋃{(Subscript[ϵ, 4])->1,(Subscript[ϵ, 6])->(Subscript[ϵ, 1])})

Solve[(1-(Subscript[ϵ, 1])+2 (Subscript[ϵ, 2]))==0∧(1+(Subscript[ϵ, 1])-2 (1+(Subscript[ϵ, 2])))==0]

Jacobi@@{a[1,1,2],a[1,1,3],a[1,2,1]} /. (ϵrules=ϵrules⋃{(Subscript[ϵ, 2])->-(1/2)+((Subscript[ϵ, 1])/2)}) // S

Jacobi@@{a[1,1,2],a[1,1,3],a[1,3,1]} /. ϵrules// S

ϵrules

{(Subscript[ϵ, 2])=-(1/2)+((Subscript[ϵ, 1])/2),(Subscript[ϵ, 4])=1,(Subscript[ϵ, 5])=1,(Subscript[ϵ, 6])=(Subscript[ϵ, 1]),(Subscript[ϵ, 7])=-(Subscript[ϵ, 6])};

{x1,x2,x3}={a[1,1,2],a[1,1,3],a[1,3,1]}

S[{B[x1,B[x2,x3]] , B[x2,B[x3,x1]] , B[x3,B[x1,x2]]}]

B[x1,B[x2,x3]] //S

VS=False;
JacErrors=DeleteCases[
bas1=FormalBasis[4,f];
bas2=FormalBasis[4,g];
bas3=FormalBasis[4,h];
Flatten[
Table[Jacobi[bas1[[i]],bas2[[j]],bas3[[k]]],
{i,Length[bas1]-1}, {j, i+1,Length@bas2}, {k,i+1,Length@bas3}]
],
0]

(* was 1737 *)Length@JacErrors

JE2=DeleteCases[JacErrors /. {(Subscript[ϵ, 6])->1}, _->0]

Length@JE2

JE2[[1]] //S

Sort[{i,j,jj,k}]