  We prove a version of the Ando-Choi-Effros lifting theorem respecting
subspaces, which in turn relies on Oja's principle of local reflexivity
respecting subspaces. To achieve this, we first develop a theory of pairs of
$M$-ideals. As a first consequence we get a version respecting subspaces of the
Michael-Pe{\l}czy\'nski extension theorem. Other applications are related to
linear and Lipschitz bounded approximation properties for a pair consisting of
a Banach space and a subspace. We show that in the separable case, the BAP for
such a pair is equivalent to the simultaneous splitting of an associated pair
of short exact sequences given by a construction of Lusky. We define a
Lipschitz version of the BAP for pairs, and study its relationship to the
(linear) BAP for pairs. The two properties are not equivalent in general, but
they are when the pair has an additional Lipschitz-lifting property in the
style of Godefroy and Kalton. We also characterize, in the separable case,
those pairs of a metric space and a subset whose corresponding pair of
Lipschitz-free spaces has the BAP.
