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