$\DeclareMathOperator\SO{SO}$At the very begining of [Akbulut and Kalafat - Algebraic topology of $G\_2$ manifolds](https://arxiv.org/abs/1308.2263v2), the authors stated that there is a "canonical fibration" for $G\_2$ of the form
$$G\_2\to \SO(7)\to \mathbb{R}P^7,$$
where the map $G\_2\to \SO(7)$ is obtained by regarding $G\_2$ as the automorphisms of the imaginary octonions, which, as a real vector space, is of dimension $7$.


So, how does one construct a homotopy equivalence $\SO(7)/G\_2\to\mathbb{R}P^7$?



The "canonical fibration" for the Lie group $G_2$
In *Spinors and Calibrations* by F. Reese Harvey, you can find proof (p. 283) of $S^7 \simeq Spin(7)/G\_2.$ It takes the same approach as Bryant's notes mentioned in the comments but it is much more detailed in this case.


The main idea is to write down the spinor representation of $Spin(7)$ using octonionic multiplication. It takes some work to see that the stabilizer of a point is then isomorphic to the automorphism group of the octonions, which is the compact Lie group $G\_2.$


