The Magic Square and Symmetric Compositions Ii

Abstract. The construction of Freudenthal’s Magic Square, which contains the exceptional simple Lie algebras of types F4, E6, E7 and E8, 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 sl2. 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 sl2 and of its natural module.