  The truncation formulae of $D=11$ supergravity on $S^7$ to $D=4$ ${\cal N}
=8$ SO(8)-gauged supergravity are completed to include the full non-linear
dependence of the $D=11$ three-form potential $\hat A_{(3)}$ on the $D=4$
fields, and their consistency is shown. The full embedding into $\hat A_{(3)}$
is naturally expressed in terms of a restricted version, still ${\cal N} =8$
but only SL(8)--covariant, of the $D=4$ tensor hierarchy. The redundancies
introduced by this approach are removed at the level of the field strength
$\hat F_{(4)}$ by exploiting $D=4$ duality relations. Finally, new expressions
for the full consistent truncation formulae are given that are, for the first
time, explicit in all $D=4$ fields.
