Automorphisms of triality groups over local rings

Abstract

Let $R$ be a nonzero commutative local ring with $2,3\in R^\times$, and let $ρ\in\Aut(R)$ satisfy $ρ^3=\id$. We classify abstract automorphisms of the elementary and full triality groups in the adjoint and simply connected forms of type $D_4$. Every automorphism is uniquely determined by an adjoint group element and a ring automorphism preserving the trace $T=\id+ρ+ρ^2$; conversely, every such pair defines an automorphism. A square-zero example shows that trace-preserving coefficient automorphisms need not normalize $\langleρ\rangle$ when $ρ$ is nontrivial on $R$ but trivial on its residue field. We also prove a relative rigidity theorem for homomorphisms from elementary triality groups over arbitrary coefficient rings to local targets with $2$ and $3$ invertible, assuming that an adjoint conjugation makes the parameter-one root images standard modulo the maximal ideal. Conversely, every unital trace-preserving ring homomorphism into such a local coefficient ring induces the corresponding homomorphism of elementary groups.