\(\textrm{Spin}(7)\) Is Unacceptable

We classify to some extent the pairs of group morphisms \(\Gamma \rightarrow \textrm{Spin}(7)\) $\mathrm{\Gamma }\to \text{Spin}(7)$ which are element-conjugate but not globally conjugate. As an application, we study the case where \(\Gamma \) $\mathrm{\Gamma }$ is the Weil group of a p-adic local field, which is relevant to the recent approach to the local Langlands correspondence for \(\textrm{G}_2\) ${\text{G}}_2$ and \(\textrm{PGSp}_6\) ${\text{PGSp}}_6$ in Gan and Savin (Forum Math Pi 11:e28, 2023). As a second application, we improve some result in Kret and Shin (J Eur Math Soc 25(1):75–152, 2023) about \(\textrm{GSpin}_7\) ${\text{GSpin}}_7$ -valued Galois representations.