Category of real Spin(8) representations

Fix the compact simply connected group $G=\operatorname{Spin}(8)$ . The category of finite-dimensional real $G$ -representations has pairs $(V,\rho)$ , with $V$ a finite-dimensional real vector space and $\rho:G\to GL(V)$ a smooth group homomorphism, as objects. A morphism $(V,\rho)\to(W,\sigma)$ is a real-linear map $T$ satisfying

T\rho(g)=\sigma(g)T\qquad(g\in G).

Identities and composition are ordinary linear identities and composition.

Triality changes the action

The vector and two half-spin representations are three different objects of this category. Their eight-dimensional carriers are isomorphic as real vector spaces, but the actions are not pairwise intertwined by an invertible map. Triality permutes them by precomposing their actions with suitable automorphisms of the group; it does not identify them as representations of the fixed group with fixed action.