  Bryant and Salamon gave a construction of metrics of G2 holonomy on the total
space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional
self-dual Einstein manifold. We generalise it by considering the total space of
an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection
on this bundle. We make essentially the same ansatz for the calibrating 3-form,
but use the curvature 2-forms instead of the ASD ones. We show that the
resulting 3-form defines a metric of G2 holonomy if the connection satisfies a
certain second-order PDE. This is exactly the same PDE that arises as the field
equation of a certain 4-dimensional gravity theory formulated as a
diffeomorphism-invariant theory of SO(3) connections. Thus, every solution of
this 4-dimensional gravity theory can be lifted to a G2-holonomy metric. Unlike
all previously known constructions, the theory that we lift to 7 dimensions is
not topological. Thus, our construction should give rise to many new metrics of
G2 holonomy. We describe several examples that are of cohomogeneity one on the
base.
