There is an evident analogy between (1) closed $G_2$structure and (2) supergravity superspace constraints. I am wondering if and where in the literature this analogy has been expanded on.
What I mean is this:
(1) On the 7dimensional Euclidean space $\mathbb{R}^7$ there is a canonical constant 3form $\phi$, often called the "associative 3form", and a $G_2$structure on a 7manifold $X$ is a 3form on $X$ which locally looks like this $\phi$.
(2) On the superMinkowski spacetime $\mathbb{R}^{10,132}$ there is a canonical constant super 4form whose components are $\Gamma_{a b \alpha \beta}$, and an onshell background of 11dimensional supergravity is constrained to have super 4form flux that locally looks like this.
In stating it this way, I am intentionally glossing over some fine print, but not over much.
For the first statement any classical source will do, a good discussion is (see in particular on p. 21) in

Robert Bryant, Some remarks on G2structures, Proceedings of the 12th Gökova GeometryTopology Conference 2005, pp. 75109 pdf
The second statement originates around
see also for instance section 3.1 of

Gianguido Dall’Agata, Davide Fabbri, Christophe Fraser, Pietro Fré, Piet Termonia, Mario Trigiante,The Osp(84) singleton action from the supermembrane, Nucl.Phys.B542:157194,1999, (arXiv:hepth/9807115)
This is for 11d supergravity. For 10d supergravity theories similar statements hold, but the formulas have more components and are a little bit less directly analogous to $G_2$structures.
And of course there is the whole story of compactifying 11d sugra on $G_2$manifolds. So what I am after here must be well known and this question will just show my ignorance of the literature, but anyway:
Where is an explicit discussion that makes the above anaology manifest, that the associative 3form on $\mathbb{R}^7$ is a local model for a differential 3form on a curved $G_2$manifold in much the same way that the super4form with compoments $\Gamma_{a b \alpha \beta}$ on $\mathbb{R}^{10,132}$ is the local model for the super4form flux on curved solutions to 11dimensional supergravity?