The GraviGUT Algebra Is not a Subalgebra of $E_8$, but $E_8$ Does Contain an Extended GraviGUT Algebra

TL;DR

This paper clarifies the definition of the GraviGUT algebra, proves it cannot embed into any real form of E8, and constructs embeddings of an extended version into E8, classifying them up to automorphism.

Contribution

It resolves ambiguities in the GraviGUT algebra's definition, proves non-embeddability into E8, and constructs and classifies embeddings of an extended algebra.

Findings

The GraviGUT algebra cannot be embedded into any real form of E8.

A modified construction yields embeddings of the extended GraviGUT algebra into E8.

Classified embeddings up to inner automorphism.

Abstract

The (real) GraviGUT algebra is an extension of the $\mathfrak{spin}(11,3)$ algebra by a $64$-dimensional Lie algebra, but there is some ambiguity in the literature about its definition. Recently, Lisi constructed an embedding of the GraviGUT algebra into the quaternionic real form of $E_8$. We clarify the definition, showing that there is only one possibility, and then prove that the GraviGUT algebra cannot be embedded into any real form of $E_8$. We then modify Lisi's construction to create true Lie algebra embeddings of the extended GraviGUT algebra into $E_8$. We classify these embeddings up to inner automorphism.