According to Bryant's work on special holonomy groups, $G_2\subset SO(7)$ may be defined as the group preserving the following 3-form:

$\phi_0=\mathrm{d}x_{123}+\mathrm{d}x_{145}+\mathrm{d}x_{167}+\mathrm{d}x_{246}-\mathrm{d}x_{257}-\mathrm{d}x_{347}-\mathrm{d}x_{356}$

where $\mathrm{d}x_{ijk}=\mathrm{d}x_i\wedge\mathrm{d}x_j\wedge\mathrm{d}x_k$. In a similar fashion, $Spin(7)\subset SO(8)$ may be defined as the group preserving the following 4-form:

$\Omega_0=\mathrm{d}x_{1234}+\mathrm{d}x_{1256}+\mathrm{d}x_{1278}+\mathrm{d}x_{1357}-\mathrm{d}x_{1368}-\mathrm{d}x_{1458}-\mathrm{d}x_{1467}-\mathrm{d}x_{2358}-\mathrm{d}x_{2367}-\mathrm{d}x_{2457}+\mathrm{d}x_{2468}+\mathrm{d}x_{3456}+\mathrm{d}x_{3478}+\mathrm{d}x_{5678}$

**But of course, with this definition, it is not obvious at all that $Spin(7)$ is really a double cover of $SO(7)$. Is there any way to understand this fact?** (that is, which is the point of defining $\Omega_0$ like this?) In the book On Quaternions and Octonions there is a definition of $Spin(7)\subset SO(8)$ related to 'octonions automorphisms' (they are not really automorphisms) which reveals some of this double-cover nature. Perhaps these could be a bridge linking these two sides, but at this moment it is not clear for me. Any suggestion will be welcomed.