Consistent $\mathcal{N}=4$, $D=4$ truncation of type IIB supergravity on ${\text{S}}^1\times {\text{S}}^5$

Fetching techniques from Generalised Geometry and Exceptional Field Theory, we develop a new method to identify consistent subsectors of four-dimensional gauged maximal supergravities that possess a (locally) geometric embedding in type IIB or 11D supergravity. We show that a subsector that is invariant under a structure group ${\text{G}}_{\text{S}}\subset {\text{E}}_{7(7)}$ can define a consistent truncation, even when ${\text{G}}_{\text{S}}$ is not part of the symmetry of the gauged maximal supergravity. As an illustration of the method, type IIB supergravity on ${\text{S}}^1\times {\text{S}}^5$ is shown to admit a consistent truncation to pure $\mathcal{N}=4$, $D=4$ gauged supergravity. Explicit uplift formulae are presented which provide a type IIB alternative to the M-theory embedding constructed by Cvetic, Lu and Pope $25$ years ago.