How to uplift D=3 maximal supergravities

We prove necessary and sufficient algebraic conditions to determine whether a D=3 gauged maximal supergravity can be obtained from consistent Kaluza-Klein truncation of ten- or eleven-dimensional supergravity. We describe the procedure to identify the internal geometry and explicitly construct the frame encoding the reduction ansatz. As byproducts, we derive several results on twistings, deformations and global aspects of E$_8$ exceptional geometry and define E$_8$ generalised diffeomorphism for massive IIA supergravity. We devise simple algebraic conditions for imposing compactness of the internal space and derive no-go results for the uplift of compact gaugings and of a large class of gauged maximal supergravities with N=(8,0) AdS$_3$ solutions.