# Rational Conjugacy Classes of Maximal Tori In Groups of Type $D_n$

## TL;DR

This paper characterizes rational conjugacy classes of maximal tori in groups of type D_n, focusing on number and p-adic fields, including various forms and Spin groups, especially for D_4.

## Contribution

It provides a concrete description of rational conjugacy classes of maximal tori in type D_n groups, covering all forms and Spin groups, including D_4.

## Key findings

- Complete classification of rational conjugacy classes in type D_n groups.
- Explicit descriptions for forms over number and p-adic fields.
- Analysis of outer forms and Spin groups, including D_4.

## Abstract

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type $D_n$, with specific emphasis on the case of number fields and p-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work we handle all (simply connected) outer forms of $D_4$.