  Vacuum multidimensional cosmological models with internal spaces being
compact $n$-dimensional Lie group manifolds are considered. Products of
3-spheres and $SU(3)$ manifold (a novelty in cosmology) are studied. It turns
out that the dynamical evolution of the internal space drives an accelerated
expansion of the external world (power law inflation). This generic solution
(attractor in a phase space) is determined by the Lie group space without any
fine tuning or arbitrary inflaton potentials. Matter in the four dimensions
appears in the form of a number of scalar fields representing anisotropic scale
factors for the internal space. Along the attractor solution the volume of the
internal space grows logarithmically in time. This simple and natural model
should be completed by mechanisms terminating the inflationary evolution and
transforming the geometric scalar fields into ordinary particles.
