Let \(E=\mathbb R^8\) with its Euclidean inner product, and let
\(\pi :G\to SO(E)\) be the vector action. On three copies of this
same real space define the twisted modules
\[V_i=(E,\pi \circ \tau ^i),\qquad i\in \mathbb Z/3\mathbb Z.\]
Their complexifications have classes \(V,S^+,S^-\), respectively,
by Theorem [fcap-003T]. Thus Proposition [fcap-003D] gives a one-dimensional
complex space of invariant trilinears on their complexifications.
The invariance equations on \(E^{\otimes 3}\) have real
coefficients. The real and imaginary parts of every invariant complex
trilinear are therefore invariant real trilinears. It follows that
\[L=(V_0^*\otimes _{\mathbb R}V_1^*\otimes _{\mathbb R}V_2^*)^G,
\qquad \dim _{\mathbb R}L=1,\]
because complexification identifies \(L\otimes _{\mathbb R}\mathbb C\)
with the invariant complex space. Choose \(0\ne t\in L\).
Write
\[D(g)=\operatorname {diag}\bigl (\pi (g),\pi (\tau g),
\pi (\tau ^2g)\bigr ).\]
Define \(Ct(x_0,x_1,x_2)=t(x_1,x_2,x_0)\) using the common
underlying space \(E\). Permuting \(D(g)(x_0,x_1,x_2)\) this way
gives \(D(\tau g)(x_1,x_2,x_0)\), so \(C\) preserves \(L\).
Moreover \(C^3=1\). On a real one-dimensional space this means
\(C=c\) for a real scalar with \(c^3=1\); hence \(c=1\). In particular,
\[t(x_0,x_1,x_2)=t(x_1,x_2,x_0)=t(x_2,x_0,x_1).\]
The real field is essential here: over \(\mathbb C\), a cube-one
scalar need not be one.
Choose complex \(G\)-module isomorphisms
\[\phi _0=1:V_{0,\mathbb C}\to V,\qquad
\phi _1:V_{1,\mathbb C}\to S^+,\qquad
\phi _2:V_{2,\mathbb C}\to S^-.\]
They exist by the proved representation comparisons. The pullback of
\(F\) along these maps is \(\lambda t_{\mathbb C}\) for some
\(\lambda \ne 0\). Multiplying \(\phi _1\) by a unit complex scalar
makes \(\lambda \) real. Now define
\(S^+_{\mathbb R}=\phi _1(E)\) and
\(S^-_{\mathbb R}=\phi _2(E)\), with inner products transported from
\(E\). They are \(G\)-invariant real forms with positive inner products,
and the restriction of \(F\) to these three real forms is real-valued.
Set
\[J_0=\phi _1\phi _0^{-1},\qquad
J_1=\phi _2\phi _1^{-1},\qquad
J_2=\phi _0\phi _2^{-1}.\]
In these formulas the underlying copies of \(E\) are identified;
the maps relate differently twisted actions. Each is a real isometry
for the stated metrics, and each cyclic composite of all three is the
identity. With \(\Phi =\phi _0\oplus \phi _1\oplus \phi _2\), let
\(J=\Phi J_E\Phi ^{-1}\), where
\(J_E(x_0,x_1,x_2)=(x_2,x_0,x_1)\). Thus \(J^3=1\).
The diagonal action on \(V\oplus S^+\oplus S^-\) is
\(\mathcal D(g)=\Phi D(g)\Phi ^{-1}\). In the common coordinates,
\[J_E D(g)=D(\tau ^{-1}g)J_E,\]
so \(J\mathcal D(g)=\mathcal D(\tau ^{-1}g)J\). Finally the cyclic
identity for \(t\) gives
\[\begin {aligned}
F(J_2s_-,J_0v,J_1s_+)
&=\lambda t_{\mathbb C}(x_2,x_0,x_1)\\
&=\lambda t_{\mathbb C}(x_0,x_1,x_2)
=F(v,s_+,s_-),
\end {aligned}\]
where \(v=\phi _0x_0\), \(s_+=\phi _1x_1\), and \(s_-=\phi _2x_2\).
No rescaling of \(F\) was made, so its top-form normalization in
Proposition [fcap-003D] is retained.