Killing Forms of Isotropic Lie Algebras

TL;DR

This paper introduces a new method for computing the Killing form of isotropic Lie algebras over arbitrary fields, simplifying calculations for complex algebra types and providing unified formulas for inner type E6 algebras.

Contribution

It presents a novel approach to compute Killing forms using subalgebras, enabling streamlined formulas for types E6 and E7, and unifies formulas for all inner type E6 Lie algebras.

Findings

Streamlined formulas for types E6 and E7

Unified formula for all inner type E6 Lie algebras

Method applicable over arbitrary fields

Abstract

This paper presents a method for computing the Killing form of an isotropic Lie algebra defined over an arbitrary field based on the Killing form of a subalgebra containing its anisotropic kernel. This approach allows for streamlined formulas for many Lie algebras of types E6 and E7 and yields a unified formula for all Lie algebras of inner type E6, including the anisotropic ones.