  Maximal supergravity in four dimensions admits two inequivalent dyonic
gaugings of the group $\mathrm{SO}(4) \times \mathrm{SO}(2,2) \ltimes T^{16}$.
Both admit a Minkowski vacuum with residual $\mathrm{SO}(4) \times
\mathrm{SO}(2)^2$ symmetry and identical spectrum. We explore these vacua and
their deformations. Using exceptional field theory, we show that the
four-dimensional theories arise as consistent truncations from IIA and IIB
supergravity, respectively, around a $\mathrm{Mink}_4\times S^3\times H^3$
geometry. The IIA/IIB truncations are efficiently related by an outer
automorphism of $\mathrm{SL}(4) \subset \mathrm{E}_{7(7)}$. As an application,
we give an explicit uplift of the moduli of the vacua into a 4-parameter family
of ten-dimensional solutions.

High Energy Physics - Theory;General Relativity and Quantum Cosmology