Embeddings of the Standard Model in $E_8$

I present a modified version of the Manogue-Dray-Wilson `octions' model of elementary particles, that overcomes some of the objections to that model that have been raised. In particular, I restore the compactness of the Standard Model gauge group, and show how the symmetry-breaking of the weak $SU(2)$ relates to the symmetry-breaking between the three generations of elementary fermions. In the process of attempting to implement a Dirac equation for three generations of fermions simultaneously, it turns out that some parts of the $E_8$ model are not required for the Standard Model, which is entirely contained in the subalgebra $\mathfrak{so}(7,3)$. In particular a re-interpretation of the Dirac spinors allows us to interpret part of the model as quantum gravity, which is then compared to General Relativity. The general structure of the model shows that the mixing angles depend on masses, and that the masses emerge from quantum interactions with the dynamic background spacetime (vacuum). Some sample calculations are given to support these predictions.