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 $F_{4,R},E_{6,R},E_{7,R},E_{8,R}$

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.