  In quantum information, lifting is a systematic procedure that can be used to
derive---when provided with a seed Bell inequality---other Bell inequalities
applicable in more complicated Bell scenarios. It is known that the procedure
of lifting introduced by Pironio [J. Math. Phys. A 46, 062112 (2005)] preserves
the facet-defining property of a Bell inequality. Lifted Bell inequalities
therefore represent a broad class of Bell inequalities that can be found in
{\em all} Bell scenarios. Here, we show that the maximal value of {\em any}
lifted Bell inequality is preserved for both the set of nonsignaling
correlations and quantum correlations. Despite the degeneracy in the maximizers
of such inequalities, we show that the ability to self-test a quantum state is
preserved under these lifting operations. In addition, except for
outcome-lifting, local measurements that are self-testable using the original
Bell inequality---despite the degeneracy---can also be self-tested using {\em
any} lifted Bell inequality derived therefrom. While it is not possible to
self-test {\em all} the positive-operator-valued measure elements using an
outcome-lifted Bell inequality, we show that partial, but robust self-testing
statements on the underlying measurements can nonetheless be made from the
quantum violation of these lifted inequalities. We also highlight the
implication of our observations on the usefulness of using lifted Bell-like
inequalities as a device-independent witnesses for entanglement depth. The
impact of the aforementioned degeneracy on the geometry of the quantum set of
correlations is briefly discussed.
