VS=True;

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

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]}

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

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]}

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

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

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 180 for FormalBasis[3], 2156 for FormalBasis[4]*)
Length@JacErrors