  Attributed to J F Adams is the conjecture that, at odd primes, the mod-p
cohomology ring of the classifying space of a connected compact Lie group is
detected by its elementary abelian p-subgroups. In this note we rely on Toda's
calculation of H^*(BF_4;F_3) in order to show that the conjecture holds in case
of the exceptional Lie group F_4. To this aim we use invariant theory in order
to identify parts of H^*(BF_4;F_3) with invariant subrings in the cohomology of
elementary abelian 3-subgroups of F_4. These subgroups themselves are
identified via the Steenrod algebra action on H^*(BF_4;F_3).
