The Magic Square and Symmetric Compositions II

Abstract

The construction of Freudenthal's Magic Square, which contains the exceptional simple Lie algebras of types $F_4,E_6,E_7$ and $E_8$, in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric composition algebras, will be defined, in the split case, in terms of the natural two dimensional module for the simple Lie algebra ${\mathfrak{sl}}_2$. As a consequence, it will be shown how all the Lie algebras in Freudenthal's Magic Square can be constructed, in a unified way, using copies of ${\mathfrak{sl}}_2$ and of its natural module.