  We characterise Lie groups with bi-invariant bargmannian, galilean or
carrollian structures. Localising at the identity, we show that Lie algebras
with ad-invariant bargmannian, carrollian or galilean structures are actually
determined by the same data: a metric Lie algebra with a skew-symmetric
derivation. This is the same data defining a one-dimensional double extension
of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with
such double extensions, containing carrollian Lie algebras as an ideal and
projecting to galilean Lie algebras. This sets up a canonical correspondence
between carrollian and galilean Lie algebras mediated by bargmannian Lie
algebras. This reformulation allows us to use the structure theory of metric
Lie algebras to give a list of bargmannian, carrollian and galilean Lie
algebras in the positive-semidefinite case. We also characterise Lie groups
admitting a bi-invariant (ambient) leibnizian structure. Leibnizian Lie
algebras extend the class of bargmannian Lie algebras and also set up a
non-canonical correspondence between carrollian and galilean Lie algebras.
