Triality rotation of connections and Dirac operators on SU(3)

We show how triality, the symmetry permuting the three 8-dimensional representations of $\mathrm{Spin}(8)$, can also appear as a symmetry of geometric objects such as connections and Dirac operators. Take any Riemannian metric $g$ on the manifold $M = \mathrm{SU}(3)$ and let $\nabla$ be a metric con