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, 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, 6])->(Subscript[ϵ, 1])})

Solve[(-1+(Subscript[ϵ, 1])-2 (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

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

ϵrules

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

{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

Jacobi@@{a[f[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])],1,2],a[g[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])],1,3],a[h[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])],3,1]}

Jacobi@@{a[1,1,2],a[1,1,3],ca[1,1,3,1]}

Jacobi@@{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],δaa[h[(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])],3,1,3,1]}

Jacobi@@{a[1,1,2],a[1,2,1],a[1,3,1]}

VS=False;
JacErrors=DeleteCases[
bas1=FormalBasis[3,f];
bas2=FormalBasis[3,g];
bas3=FormalBasis[3,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 186 for FormalBasis[3], 2156 for FormalBasis[4]*)
Length@JacErrors