The Allison-Faulkner construction of $E_8$

TL;DR

This paper investigates the limitations of the Tits construction for the exceptional Lie algebra $E_8$, introducing cohomological invariants to distinguish between different algebraic constructions.

Contribution

It demonstrates that the Tits index $E_8^{133}$ cannot be realized over fields without odd degree extensions and develops new invariants to analyze the isotropy rank of $E_8$ constructions.

Findings

The Tits index $E_8^{133}$ is not obtainable over certain fields.

Two cohomological invariants in degrees 6 and 8 are constructed.

Invariants can detect the isotropy rank of $E_8$ constructions.

Abstract

We show that the Tits index $E_8^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. We construct two cohomological invariants, in degrees 6 and 8, of the Tits construction and the more symmetric Allison-Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.