  Peering in from the outside, $\mathbb{A} :=
\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ looks to be an
ideal mathematical structure for particle physics. It is 32
$\mathbb{C}$-dimensional: exactly the size of one full generation of fermions.
Furthermore, as alluded to earlier in arXiv:1806.00612, it supplies a richer
algebraic structure, which can be used, for example, to replace SU(5) with the
SU(3)$\times$SU(2)$\times$U(1) / $\mathbb{Z}_6$ symmetry of the standard model.
However, this line of research has been weighted down by a difficulty known as
the fermion doubling problem. That is, a satisfactory description of
SL(2,$\mathbb{C}$) symmetries has so far only been achieved by taking two
copies of the algebra, instead of one. Arguably, this doubling of states
betrays much of $\mathbb{A}$'s original appeal. In this article, we solve the
fermion doubling problem in the context of $\mathbb{A}$. Furthermore, we give
an explicit description of the standard model symmetries, $g_{sm}$, its gauge
bosons, Higgs, and a generation of fermions, each in the compact language of
this 32 $\mathbb{C}$-dimensional algebra. Most importantly, we seek out the
subalgebra of $g_{sm}$ that is invariant under the complex conjugate - and find
that it is given by $su(3)_C \oplus u(1)_{EM}$. Could this new result provide a
clue as to why the standard model symmetries break in the way that they do?

High Energy Physics - Phenomenology;High Energy Physics - Theory