The octonions $\mathbb{O}$ are the $8$-dimension real (non-associative) normed division algebra. Forgetting the algebra structure leads to an identification of $\mathbb{O}$ with $\mathbb{R}^8$ as real normed spaces, and so generates a tautological representation of $SO(8)$ on $\mathbb{O}$.
Now one may define the compact exceptional group $G_2$ as the $\mathbb{R}$-linear algebra automorphism group of $\mathbb{O}$. That is,
$$G_2=\{\alpha\in SO(8)\mid \alpha(xy)=\alpha(x)\alpha(y),\,\forall\, x,y\in\mathbb{O}\}\leq SO(8).$$
On the other hand we can appeal to Cartan's principle of triality in $SO(8)$ and define a subgroup of $SO(8)$ which identifies with a double cover of $SO(7)$. In short, we have
$$Spin_7\cong\{\alpha\in SO(8)\mid\exists \tilde\alpha\in SO(7)\,\text{such that}\,\alpha(xy)=\tilde\alpha(x)\alpha(y)\,\forall\,x,y\in\mathbb{O}\}\leq SO(8).$$
Then clearly $G_2\leq Spin_7$. Moreover this $G_2$ subgroup preserves the algebra unit $1\in\mathbb{O}$, and if we let $Spin_7$ act on the unit sphere $S^7\subseteq \mathbb{O}$ through its inclusion in $SO(8)$ we are led to a fibre sequence
$$G_2\rightarrow Spin_7\rightarrow S^7.$$
Now this fibration is principal, classified by a map $\gamma:S^7\rightarrow BG_2$ where $BG_2$ is the classifying space of $G_2$. If we check the homotopy groups of $G_2$ we find that $\pi_7BG_2\cong\pi_6G_2\cong\mathbb{Z}_3$ and it turns out that $\gamma$ is a generator for the first group. On the other hand, if we localise these spaces away from $3$, then clearly $(\pi_7BG_2)_{(\frac{1}{3})}=0$ and (the localisation of) $\gamma$ is null-homotopic. In this case the fibration above splits, and we get a $\frac{1}{3}$-local homotopy equivalence
$$Spin_7\simeq_{\frac{1}{3}}S^7\times G_2,$$
although one should be quick to remark that this homotopy equivalence is one of spaces, not $H$-spaces.
This brings me to my actual question.
Is there an algebraic or geometric explanation for why the $\frac{1}{3}$-local homotopy equivalence $Spin_7\simeq_{\frac{1}{3}}S^7\times G_2$ should exist?
The splitting is clearly a homotopy-theoretic artifact, rather than directly a geometric or algebraic consequence. So perhaps the question should rather be, is there an algebraic or geometric reason why the homotopy-theoretic data aligns itself in this way?
As an example of what I'm looking for, the exceptional isomorphism $Spin_4\cong S^3\times S^3$ can be very explicitly seen by writing down a simple representation of $Spin_4$ and determining the consequences. It seems that the explanation for the homotopy equivalence I am considering, however, would be much more subtle.