Proposition 3.3 (Polarisation identity for symmetric (4,0)-tensors).

eta1={A1,A2,A3,A4};
eta2={B1,B2,B3,B4};
eta3={C1,C2,C3,C4};
eta4={D1,D2,D3,D4};

s[x_]:=(-1)^x

arg[th1_, th2_, th3_,th4_]:= s[th1] eta1 + s[th2]eta2+s[th3]eta3+s[th4]eta4

Lread[a_,b_,c_,d_]:=Module[{tmp},
tmp=Sort[{a,b,c,d}];
ToExpression["Lnormal"<>ToString[tmp[[1]]]<>ToString[tmp[[2]]]<>ToString[tmp[[3]]]<>ToString[tmp[[4]]]]
]

L[x1_,x2_,x3_,x4_]:=Sum[
Lread[a,b,c,d] x1[[a]] x2[[b]] x3[[c]] x4[[d]],
{a,1,4},{b,1,4},{c,1,4},{d,1,4}
]

RHS=Sum[s[th1] s[th2] s[th3]s[th4] L[ arg[th1,th2,th3,th4],arg[th1,th2,th3,th4],arg[th1,th2,th3,th4],arg[th1,th2,th3,th4]],
{th1,0,1},{th2,0,1},{th3,0,1},{th4,0,1}
];
RHS=Expand[1/(4! 2^4)RHS];

Expand[RHS-L[eta1,eta2,eta3,eta4]]

4! 2^4

Extra

RHS