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](http://rads.stackoverflow.com/amzn/click/1568811349) 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.



$Spin(7)$ as stabilizer of a $4$-form
There are at least two sources for this: First, Harvey and Lawson, *Calibrated geometries* (Acta Math. 1982) proves this (i.e., that the stabilizer of $\Omega\_0$ is isomorphic to the nontrivial double cover of $\mathrm{SO}(7)$) using properties of the octonions. Second you can find a proof that doesn't rely on knowledge of the octonions but instead relies on the classification of Lie algebras and groups in Bryant's *Metrics with exceptional holonomy* (Ann. of Math, 1987, Theorem 4 of Section 2). 


There are even more 'direct' (i.e., elementary) proofs available, of course. If those don't satisfy you, then ask again and I'll sketch one here when I have time.


