In this paper we construct a family of spin Lie groups G with an outer automorphism of order three (triality automorphism) and we describe the subgroups of fixed points for this kind of automorphisms. We will take advantage of this work to study the action of the group of outer automorphisms of G on the moduli space of principal G-bundles and describe the subvariety of fixed points in M(G) for the action of the outer automorphism of order three of G. Finally, we further study the case of Spin(8, C).