$\mathbf {5 \times 5}$ -graded Lie algebras, cubic norm structures and quadrangular algebras

Abstract

We study simple Lie algebras generated by extremal elements, over arbitrary fields of arbitrary characteristic. We show the following: (1) If the extremal geometry contains lines, then the Lie algebra admits a $5 \times 5$ -grading that can be parametrized by a cubic norm structure; (2) If there exists a field extension of degree at most $2$ such that the extremal geometry over that field extension contains lines, and in addition, there exist symplectic pairs of extremal elements, then the Lie algebra admits a $5 \times 5$ -grading that can be parametrized by a quadrangular algebra.

Keywords

• 17B05
• 17B20
• 17B25
• 17B40
• 17B45
• 17B60
• 17C40
• 17A30
• 20G15
• 20G41
• 51A50