Gradings on the exceptional Lie algebras $F_4$ and $G_2$ revisited

Abstract

All gradings by abelian groups are classified on the following algebras over an algebraically closed field $\mathbb{F}$: the simple Lie algebra of type $G_2$ (char $\mathbb{F}\ne 2,3$), the exceptional simple Jordan algebra (char $\mathbb{F}\ne 2$), and the simple Lie algebra of type $F_4$ (char $\mathbb{F} \ne 2$).