  We explore geometries that give rise to a novel algebraic structure, the
Exceptional Drinfeld Algebra, which has recently been proposed as an approach
to study generalised U-dualities, similar to the non-Abelian and Poisson-Lie
generalisations of T-duality. This algebra is generically not a Lie algebra but
a Leibniz algebra, and can be realised in exceptional generalised geometry or
exceptional field theory through a set of frame fields giving a generalised
parallelisation. We provide examples including "three-algebra geometries",
which encode the structure constants for three-algebras and in some cases give
novel uplifts for $CSO(p,q,r)$ gaugings of seven-dimensional maximal
supergravity. We also discuss the M-theoretic embedding of both non-Abelian and
Poisson-Lie T-duality.
