The GraviGUT Algebra Is not a Subalgebra of E₈, but E₈ Does Contain an Extended GraviGUT Algebra

Abstract

The (real) GraviGUT algebra is an extension of the 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 E₈. 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 E₈. We then modify Lisi's construction to create true Lie algebra embeddings of the extended GraviGUT algebra into E₈ We classify these embeddings up to inner automorphism.