Rational forms of exceptional dual pairs

Abstract

Let g be a simple Lie algebra of type F4 or En defined over a local or global field k of characteristic zero. We show that g can be obtained by the Tits construction from an octonion algebra O and a cubic Jordan algebra J. In particular, g contains a dual pair h defined over k which is the direct sum of the derivation algebras of O and J. We determine the conjugacy classes of k-forms of h in g.