How to uplift non-maximal gauged supergravities

In this paper, we provide an algorithm to perform the uplift of non-maximal $G_g$-gauged supergravities to type IIB or 11D supergravities. Using tools of exceptional field theory and generalised geometry, we show that the internal manifold admits a $G_g$-action, and that consistency of the uplift is equivalent to solving a simpler PDE on the quotient $M_{\text{int}}/G_g$. As an application, we classify all possible uplifts of pure half-maximal four-dimensional $\textrm{SO}(4)$-gauged supergravity to type IIB and we recover consistent truncations around any of the D'Hoker-Estes-Gutperle solutions \cite{DHoker:2007hhe}.