Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse the usual process of constructing the Lie algebras of types G₂ and F₄ as algebras of derivations of the split octonions or the exceptional Jordan algebra and instead begin with their Dynkin diagrams and then construct the algebras together with an action of the Lie algebras and associated Chevalley groups. This is shown to be a variation on a general construction of all standard modules for simple Lie algebras and it is well suited for use in computational algebra systems. All the structure constants which occur are integral and hence the construction specialises to all fields, without restriction on the characteristic, avoiding the usual problems with characteristics 2 and 3.