Lie algebraic Carroll/Galilei duality

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