$S_4$-symmetry on the Tits construction of exceptional Lie algebras and superalgebras

Abstract

The classical Tits construction provides models of the exceptionalsimple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group $S_4$ on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too.