Editing Grand Unified Theory (section)

=== Octonion representations ===
It can be noted that a generation of 16&nbsp;fermions can be put into the form of an [[octonion]] with each element of the octonion being an 8-vector. If the 3&nbsp;generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional (Grassmann) [[Jordan algebra]], which has the symmetry group of one of the exceptional Lie groups ({{math|F}}{{sub|4}}, {{math|E}}{{sub|6}}, {{math|E}}{{sub|7}}, or {{math|E}}{{sub|8}}) depending on the details.
: <math>
\psi=
\begin{bmatrix}
a & e & \mu \\
\overline{e} & b & \tau \\
\overline{\mu} & \overline{\tau} & c
\end{bmatrix}
</math>
: <math>\ [\psi_A,\psi_B] \subset \mathrm{J}_3(\mathbb{O})\ </math>

Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known that {{math|E}}{{sub|6}} has subgroup {{math|O(10)}} and so is big enough to include the Standard Model. An {{math|E}}{{sub|8}} gauge group, for example, would have 8&nbsp;neutral bosons, 120&nbsp;charged bosons and 120&nbsp;charged anti-bosons. To account for the 248&nbsp;fermions in the lowest multiplet of {{math|E}}{{sub|8}}, these would either have to include anti-particles (and so have [[baryogenesis]]), have new undiscovered particles, or have gravity-like ([[spin connection]]) bosons affecting elements of the particles spin direction. Each of these possesses theoretical problems.