Calibrations close to ${\mathrm{G}}_2$-twistor space

We study natural structures of the kind of ${\mathrm{G}}_2$ existing on the unit tangent sphere bundle $SM\to M$ of any oriented Riemannian 4-manifold $M$. Based on the Harvey-Morgan classification of degree 3 calibrations on ${\mathbb{R}}^7$, we identity those which can be found on $SM$. We also study the Laplacian of the Sasaki metric on 3-forms.