$\DeclareMathOperator\Spin{Spin}$I am looking for a proof of a fact (I think it's true intuitively due to representation theory) in $\Spin(7)$ geometry. Let's take a closed $\Spin(7)$-manifold $(M^8,g)$. The $\Spin(7)$-structure in induced by a four-form $\Phi$ and it's torsion-free. The three-forms split: \begin{align*} \Omega^3=\Omega^3_8\oplus \Omega^3_{48} \end{align*} and the self-dual four forms split: \begin{align*} \Omega^4_+=\Omega^4_1\oplus\Omega^4_7\oplus\Omega^4_{27}. \end{align*} Take $\beta\in\Omega^3_8,$ then I want to prove that $d\beta_+$ has no component in $\Omega^4_{27}.$ The intuition is that there isn't a canonical map from an $8$-dimensional representation to a $27$-dimensional representation. I don't know how to prove this rigorously. Any proof or reference is most welcome.
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