  It was proven by Hitchin that any solution of his evolution equations for a
half-flat SU(3)-structure on a compact six-manifold M defines an extension of M
to a seven-manifold with holonomy in G_2. We give a new proof, which does not
require the compactness of M. More generally, we prove that the evolution of
any half-flat G-structure on a six-manifold M defines an extension of M to a
Ricci-flat seven-manifold N, for any real form G of SL(3,C). If G is
noncompact, then the holonomy group of N is a subgroup of the noncompact form
G_2^* of G_2^C. Similar results are obtained for the extension of nearly
half-flat structures by nearly parallel G_2- or G_2^*-structures, as well as
for the extension of cocalibrated G_2- and G_2^*-structures by parallel
Spin(7)- and Spin(3,4)-structures, respectively. As an application, we obtain
that any six-dimensional homogeneous manifold with an invariant half-flat
structure admits a canonical extension to a seven-manifold with a parallel G_2-
or G_2^*-structure. For the group H_3 \times H_3, where H_3 is the
three-dimensional Heisenberg group, we describe all left-invariant half-flat
structures and develop a method to explicitly determine the resulting parallel
G_2- or G_2^*-structure without integrating. In particular, we construct three
eight-parameter families of metrics with holonomy equal to G_2 and G_2^*.
Moreover, we obtain a strong rigidity result for the metrics induced by a
half-flat structure (\omega,\rho) on H_3 \times H_3 satisfying \omega(Z,Z)=0
where Z denotes the centre. Finally, we describe the special geometry of the
space of stable three-forms satisfying a reality condition. Considering all
possible reality conditions, we find four different special K\"ahler manifolds
and one special para-K\"ahler manifold.
