  The folding model analysis of the elastic $\alpha + \alpha$ scattering at the
incident energies below the reaction threshold of 34.7 MeV (in the lab system)
has been done using the well-tested density dependent versions of the M3Y
interaction and realistic choices for the $^4$He density. Because the
absorption is negligible at the energies below the reaction threshold, we were
able to probe the $\alpha + \alpha$ optical potential at low energies quite
unambiguously and found that the $\alpha + \alpha$ overlap density used to
construct the density dependence of the M3Y interaction is strongly distorted
by the Pauli blocking. This result gives possible explanation of a
long-standing inconsistency of the double-folding model in its study of the
elastic $\alpha + \alpha$ and $\alpha$-nucleus scattering at low energies using
the same realistic density dependent M3Y interaction.
