  We present examples of noncommutative four-spheres that are base spaces of
$SU(2)$-principal bundles with noncommutative seven-spheres as total spaces.
The noncommutative coordinate algebras of the four-spheres are generated by the
entries of a projection which is invariant under the action of $SU(2)$. We give
conditions for the components of the Connes--Chern character of the projection
to vanish but the second (the top) one. The latter is then a non zero
Hochschild cycle that plays the role of the volume form for the noncommutative
four-spheres.
