I asked this question on stackexchange (https://math.stackexchange.com/questions/2212226/restriction-of-spin7-4-form-to-mathbbr-times-s7) but was advised to ask again here:

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.