On realizations of the complex Lie groups $ (F_{4,R})^C, (E_{6,R})^C, (E_{7,R})^C ,(E_{8,R})^C$ and those compact real forms $F_{4,R},E_{6,R},E_{7,R},E_{8,R}$

TL;DR

This paper explores alternative realizations of complex and compact real forms of exceptional Lie groups by replacing the Cayley algebra with the real numbers, aiming to understand their structures.

Contribution

It introduces new realizations of exceptional Lie groups using the real number field instead of the Cayley algebra, and determines their group structures.

Findings

New realizations of complex exceptional Lie groups using real numbers.

Structural descriptions of these groups in the new realizations.

Comparison with traditional Cayley algebra-based definitions.

Abstract

In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C, {E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually use the Cayley algebra $ \mathfrak{C} $. In the present article, we consider replacing the Cayley algebra $ \mathfrak{C} $ with the field of real numbers $\mathbb R$ in the definition of the groups above, and these groups are denoted as in title above. Our aim is to determine the structure of these groups. We call realization to determine the structure of the groups.