Sporadic Isogenies to the Quaternionic Orthogonal Groups $\operatorname{SO}^*(2n)$

Abstract

For small dimensional Lie algebra's there are many so-called accidental isomorphisms which give rise to double covers of special orthogonal groups - Spin groups - which happen to coincide with groups already belonging to another classification. The well known catalog of these sporadic isogenies given by Dr. Paul Garrett keeps track of these facts for the complex orthogonal groups $\operatorname{SO}(n, \mathbb{C})$, and those real forms which are manifestly real: $\operatorname{SO}(p,q)$, while avoiding the quaternionic real forms, $\operatorname{SO}^*(2n)$. This article serves to complement and complete the existing catalog by presenting the sporadic isogenies to the first four quaternionic orthogonal groups in one place, with contemporary proofs of those group and algebra homomorphisms. A review of the definition of $\operatorname{SO}^*(2n)$ is presented in a modern notation, emphasizing the concept of `quaternion reversion' as a useful, albeit redundant, second conjugation upon quaternions, helping to explicate the self-conjugacy of quaternionic representations. Lastly, the triality of $\operatorname{SO}^*(8)$ is explored in a similar manner to work done by the author previously.