From cubic norm pairs to $G_2$- and $F_4$-graded groups and Lie algebras

Abstract

We construct Lie algebras arising from cubic norm pairs over arbitrary commutative base rings. Such Lie algebras admit a grading by a root system of type $G_2$, and when the cubic norm pair is a cubic Jordan matrix algebra, the $G_2$-grading can be further refined to an $F_4$-grading. We then use these Lie algebras and their gradings to construct corresponding root graded groups. Along the way, we produce many results providing detailed information about the structure of these Lie algebras and groups.