$|3|-$gradings of complex simple Lie algebras

Abstract

The aim of this paper is to investigate the algebraic structure that appears on $|3|-$gradings $\mathfrak{n}=\mathfrak{n}_{-3}\oplus \cdots \oplus \mathfrak{n}_3$ of a complex simple Lie algebra $\mathfrak{n}$. In particular, we completely determine the possible reductive algebras $\mathfrak{n}_0$ and prove that the only free nilpotent Lie algebra of step 3 that appears as the negative part $\mathfrak{n}_{-3}\oplus\mathfrak{n}_{-2}\oplus\mathfrak{n}_{-1}$ of a grading is the usual $|3|-$grading of the exceptional Lie algebra $\mathfrak{g}_2$.