The magic square of Lie groups: the $2 \times 2$ case

TL;DR

This paper provides a unified and explicit matrix group representation of the $2 imes 2$ Freudenthal-Tits magic square of Lie groups, enhancing understanding of their structure and relationships.

Contribution

It offers a novel, explicit matrix-based construction of the $2 imes 2$ magic square, connecting Lie groups with composition algebras.

Findings

Explicit matrix group representations for the $2 imes 2$ magic square.

Unified framework linking Lie groups and composition algebras.

Enhanced understanding of the structure of these Lie groups.

Abstract

A unified treatment of the $2 \times 2$ analog of the Freudenthal-Tits magic square of Lie groups is given, providing an explicit representation in terms of matrix groups over composition algebras.