# Revista Matemática Iberoamericana

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**Volume 23, Issue 1, 2007, pp. 57–84**

**DOI: 10.4171/RMI/486**

Published online: 2007-04-30

The Magic Square and Symmetric Compositions II

Alberto Elduque^{[1]}(1) Universidad de Zaragoza, Spain

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.

*Keywords: *Freudenthal Magic Square, symmetric composition algebra, triality, exceptional Lie algebra

Elduque Alberto: The Magic Square and Symmetric Compositions II. *Rev. Mat. Iberoam.* 23 (2007), 57-84. doi: 10.4171/RMI/486