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*(BF4;F3) in order to show that
the conjecture holds in case of the exceptional Lie group F4.
To this aim we use invariant theory in order to identify parts of
H*(BF4;F3) with invariant
subrings in the cohomology of elementary abelian 3–subgroups of
F4. These subgroups themselves are identified via the Steenrod
algebra action on H*(BF4;F3).