  Up to equivalence, this paper classifies all the irreducible unitary
representations with non-zero Dirac cohomology for the following simple real
exceptional Lie groups: ${\rm EI}=E_{6(6)}, {\rm EIV}=E_{6(-26)}, {\rm
FI}=F_{4(4)}, {\rm FII}=F_{4(-20)}$. Along the way, we find an irreducible
unitary representation of $F_{4(4)}$ whose Dirac index vanishes, while its
Dirac cohomology is non-zero. This disproves a conjecture raised in 2015
asserting that there should be no cancellation between the even part and the
odd part of the Dirac cohomology.
