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

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.