# Restriction of "Spin(7) 4-form" to $\mathbb{R}_+\times S^7$

I'm currently reading through Jason Lotay's paper "Associative Submanifolds of the 7-sphere" (https://arxiv.org/pdf/1006.0361v1.pdf) and the corresponding slides. To construct a positive 3-form on $S^7$ corresponding to a $G_2$-structure he first constructs the "$Spin(7)$ 4-form" $\Phi_0$ on $\mathbb{R}^8=\mathbb{R}\oplus\mathbb{R}^7$ as

$$\Phi_0=dx_0\wedge\phi_0+\star\phi_0$$

where $\phi_0$ is the "associative 3-form on $\mathbb{R}^7$ and $\star$ is the Hodge dual. He then claims that since $\Phi_0$ is self dual, the restriction to $\mathbb{R}^8 - \{0\}=\mathbb{R}_+\times S^7$ is given by

$$\Phi=r^3dr\wedge \phi + r^4\star\phi$$

where $\phi$ is the desired 3-form on $S^7$. However, it is not immediately clear to me how he arrives at this equality.

One way of looking at this is that you know that $$\Phi_0$$ induces the standard Euclidean metric $$g_{0}$$ on $$\mathbb{R}^8$$ and that the restriction of this metric to $$S^7$$ is the round metric, i.e. $$g_{0}=dr \otimes dr + r^2 g_{S^7}.$$ Moreover the volume forms of $$\mathbb{R}^8$$ and $$S^7$$ (with the induced orientation) are related by $$vol_0=r^7 dr \wedge vol_{S^7}.$$ You can define the $$G_2$$-structure $$\phi$$ on the unit $$S^7$$ by $$\phi=\iota_{\partial_r} \Phi_0$$. The expression for $$\Phi=r^3 dr \wedge \phi + r^4 \star \phi$$ that you give now follows from this and the fact that $$\Phi$$ is self-dual i.e. $$\star_0(r^3 dr \wedge \phi)=r^4 \star \phi$$.
Note that the $$\star$$ appearing here is with respect to the round metric and volume form of $$S^7$$. However $$\Phi$$ is a self-dual 4-form with respect to the Hodge star $$\star_0$$ of the Euclidean metric and volume form of $$\mathbb{R}^8$$. The expression is essentially obtained by comparing these two Hodge stars and taking into account the scaling factor of r.