We introduce the notion of ‐algebroid, generalising both Lie and Courant algebroids, as well as the algebroids used in exceptional generalised geometry for . Focusing on the exceptional case, we prove a classification of “exact” algebroids and translate the related classification of Leibniz parallelisable spaces into a tractable algebraic problem. After discussing the general notion of Poisson–Lie duality, we show that the Poisson–Lie U‐duality is compatible with the equations of motion of supergravity.

We discuss the conditions under which non-abelian T-duality can be considered as a chain of abelian T-dualities. Motivated by these results, we propose that the topology of a non-abelian T-dual should be phrased in the language of T-folds, and give the explicit O(d, d) transformations which can be used to glue the dual space.

In this note we study exceptional algebroids, focusing on their relation to type IIB superstring theory. We show that a IIB‐exact exceptional algebroid (corresponding to the group Enfalse(nfalse)×R+$\mathsf {E}_{n(n)}\times \mathbb {R}^+$, for n≤6$n\le 6$) locally has a standard form given by the exceptional tangent bundle. We derive possible twists, given by a flat frakturglfalse(2,double-struckRfalse)$\mathfrak {gl}(2,\mathbb {R})$‐connection, a covariantly closed pair of 3‐forms, and a 5‐form, and comment on their physical interpretation. Using this analysis we reduce the search for Leibniz parallelisable spaces, and hence maximally supersymmetric consistent truncations, to a simple algebraic problem. We show that the exceptional algebroid perspective also gives a simple description of Poisson–Lie U‐duality without spectators and hence of generalised Yang–Baxter deformations.

We study the T-dualisability criteria of Chatzistavrakidis, Deser and Jonke [3] who recently used Lie algebroid gauge theories to obtain sigma models exhibiting a "Tduality without isometry". We point out that those T-dualisability criteria are not written invariantly in [3] and depend on the choice of the algebroid framing. We then show that there always exists an isometric framing for which the Lie algebroid gauging boils down to standard Yang-Mills gauging. The "T-duality without isometry" of [3] is therefore nothing but traditional isometric non-Abelian T-duality in disguise.