Possible ambient kinematics

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.