  In a seminal paper, Bacry and L\'evy-Leblond classified kinematical algebras,
a class of Lie algebras encoding the symmetries of spacetime. Homogeneous
spacetimes (infinitesimally, Klein pairs) associated to these possible
kinematics can be partitioned into four families -- riemannian, lorentzian,
galilean and carrollian -- based on the type of invariant metric structure they
admit. In this work, we classify possible ambient kinematics -- defined as
extensions of kinematical algebras by a scalar ideal -- as well as their
associated Klein pairs. Kinematical Klein pairs arising as quotient space along
the extra scalar ideal are said to admit a lift into the corresponding ambient
Klein pair. While all non-galilean Klein pairs admit a unique -- trivial and
torsion-free -- higher-dimensional lift, galilean Klein pairs are
constructively shown to admit lifts into two distinct families of ambient Klein
pairs. The first family includes the bargmann algebra as well as its
curved/torsional avatars while the second family is novel and generically
allows lifts into torsional ambient spaces. We further comment on the relation
between these two families and the maximally symmetric family of leibnizian
Klein pairs.
