  In this paper we prove the theorem that there exists no 7--dimensional Lie
group manifold G of weak G2 holonomy.
  We actually prove a stronger statement, namely that there exists no
7--dimensional Lie group with negative definite Ricci tensor Ric_{IJ}.
  This result rules out (supersymmetric and non--supersymmetric)
  Freund--Rubin solutions of M--theory of the form AdS_4\times G and
compactifications with non--trivial 4--form fluxes of Englert type on an
internal group manifold G. A particular class of such backgrounds which, by our
arguments are excluded as bulk supergravity compactifications corresponds to
the so called compactifications on twisted--tori, for which G has structure
constants $\tau^K{}_{IJ}$ with vanishing trace $\tau^J{}_{IJ}=0$. On the other
hand our result does not have bearing on warped compactifications of M--theory
to four dimensions and/or to compactifications in the presence of localized
sources (D--branes, orientifold planes and so forth). Henceforth our result
singles out the latter compactifications as the preferred hunting grounds that
need to be more systematically explored in relation with all compactification
features involving twisted tori.
