  We find a remarkable family of $\mathrm{G}_2$ structures defined on certain
principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ associated with any
given oriented Riemannian 4-manifold $M$. Such structures are always
cocalibrated. The study starts with a recast of the Singer-Thorpe equations of
4-dimensional geometry. These are applied to the Bryant-Salamon cons\-truction
of complete $\mathrm{G}_2$-holonomy metrics on the vector bundle of self- or
anti-self-dual 2-forms on $M$. We then discover new examples of that special
holonomy on disk bundles over ${\cal H}^4$ and ${\cal H}^2_{\mathbb{C}}$,
respectively, the real and complex hyperbolic space. Only in the end we present
the new $\mathrm{G}_2$ structures on principal bundles.
