On the embedding of $C_3$ in $E_8$

I investigate the structure of $E_8$ under the action of the subalgebra/subgroup $A_1+G_2+C_3$, as a potential route to unification of the fundamental forces of nature into a single algebraic structure. The particular real form $E_{8(-24)}$ supports a decomposition into compact $G_2$ plus split $A_1+C_3$, which allows a restriction from $G_2$ to $SU(3)$ for QCD, together with split $SL_2(\mathbb R)$ to break the symmetry of the weak interaction and give mass to the bosons. The factor $C_3$ contains a copy of the Lorentz group $SL_2(\mathbb C)$ and extends the `spacetime' symmetries to the full group of symplectic symmetries of $3+3$-dimensional phase space.