A family of Spin(8) dual pairs: the case of real groups

Exceptional groups of type $E_6$ contain dual pairs where one member is $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(8)$, and the other is $T⋊\mathbb{Z}\mathrm{/}2\mathbb{Z}$, where $T$ is a two-dimensional torus and the non-trivial element in $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ acts on $T$ by the inverse involution. We describe the correspondence of representations arising by restricting the minimal representation.