Let's take a $G\_2$ manifold $(M,\Phi)$, then we get a Spin$(7)$ manifold by taking $(M\times\mathbb{R},\Psi:=\Phi\wedge dt+\*\_M\Phi),$ where $t$ is the coordinate in the $\mathbb{R}$-direction. $\Phi\in\Omega^3(M),\Psi\in\Omega^4(M\times\mathbb{R})$ determining the corresponding $G\_2$ and Spin$(7)$ structures. Now these structures give us splitting of forms. E.g.,
\begin{align\*}
&\text{On } G\_2: \Lambda^4=\Lambda^4\_1\oplus \Lambda^4\_7\oplus \Lambda^4\_{27},\hspace{1 ex}\Lambda^3=\Lambda^3\_1\oplus \Lambda^3\_7\oplus \Lambda^3\_{27},\\
&\text{On Spin}(7) : \Lambda^4=\Lambda^4\_1\oplus \Lambda^4\_7\oplus \Lambda^4\_{27}\oplus \Lambda^4\_{35}
\end{align\*}
We start with a four form say, $\alpha\in\Omega^4(M\times\mathbb{R}).$ Now the component $\alpha\_7\in\Omega^4\_7(M\times\mathbb{R}).$ Moreover let's take $\alpha=\alpha\_1+\alpha\_2\wedge dt$, where $\alpha\_1\in\Omega^4(M)$ and $\alpha\_2\in\Omega^3(M)$.We can write $\alpha\_7=\gamma+\beta\wedge dt$ (at least pointwise), where $\gamma\in\Omega^4(M)$ and $\beta\in\Omega^3(M)$. My question is what are the components of $\gamma$ and $\beta$ in $\Omega^4(M)$ and $\Omega^3(M)$ in terms of the splitting coming from the $G\_2$ structure.


The question may seem a bit cryptic, I have done the part for a form in $\Omega^2\_7$. Hope this clears things up. The known descriptions for $\Omega^4\_7$ in the literature are not very calculation-friendly. That's where my difficulty arises.


On a $G\_2$ manifold $(M,\Phi)$, $\Omega^2=\Omega^2\_7\oplus \Omega^2\_{14},$ and for a $2$-form $\xi,\xi\_7=\frac{1}{3}\big(\xi+\*(\xi\wedge\Phi)\big).$  

On a Spin-$(7)$ manifold $(N,\Psi)$, $\Omega^2=\Omega^2\_7\oplus \Omega^2\_{21},$ and for a $2$-form $\xi,\xi\_7=\frac{1}{4}\big(\xi+\*(\xi\wedge\Psi)\big).$


\begin{align\*}
 &\xi\_{\Omega^2\_7(M\times\mathbb{R})}\\
 &=\frac{1}{4}\big(\xi+\*(\xi\wedge(\Phi\wedge dt+\*\_M\Phi)\big)\\
 &=\frac{1}{4}\big(\xi+\*\_M(\xi\wedge\Phi)+\*(\xi\wedge\*\_M\Phi)\big)\\
 &=\frac{3}{4} \xi\_{\Omega^2\_7(M)}+\frac{1}{4}\*\_M(\xi\wedge\*\_M\Phi)\wedge dt
\end{align\*}



Decomposition of forms on a Spin$(7)$ manifold
Observe that, in a $\mathrm{Spin}(7)$-manifold, since $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$, the $\mathrm{Spin}(7)$ decomposition $\Lambda^4 = \Lambda^4\_1\oplus \Lambda^4\_7\oplus\Lambda^4\_{27}\oplus\Lambda^4\_{35}$ must refine the splitting $\Lambda^4=\Lambda^4\_{+}\oplus\Lambda^4\_{-}$ into self-dual and anti-self-dual parts. Since $\Lambda^4\_1$ is self-dual, it follows that
$\Lambda^4\_{+} = \Lambda^4\_1\oplus \Lambda^4\_7\oplus\Lambda^4\_{27}$ and
$\Lambda^4\_{-} = \Lambda^4\_{35}$.


Thus, $\Lambda^4\_7$ consists of self-dual forms, and hence, under the $G\_2$-splitting $\alpha\_7 = \beta\wedge \mathrm{d}t + \gamma$, since $G\_2$ acts irreducibly on the $7$-dimensional representation, we must have $\beta\in\Lambda^3\_7(M)$ and $\gamma = -{\ast}\beta\in \Lambda^4\_7(M)$.


