  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 the strong
force, together with split $SL_2(\mathbb R)$ to break the symmetry of the weak
interaction and give mass to the intermediate vector bosons. The factor $C_3$
contains various copies of the Lorentz group $SL_2(\mathbb C)$ and extends the
`spacetime' symmetries to the full group of symplectic symmetries of real
$3+3$-dimensional phase space.
  Restricting $G_2$ to the Standard Model $SU(3)$ extends $C_3$ to $A_5$, in
the real form $SU(3,3)$, acting on a complex phase space that includes both
momentum and current. There is then a natural restriction from $SU(3,3)$ to
$SO(3,3)$, describing the action of $SL_4(\mathbb R)$ on phase space. The
resulting action of $SL_4(\mathbb R)$ on $E_8$ includes tensors that are
equivalent to the stress-energy tensor, the Ricci tensor and the Riemann
tensor, and therefore permits the formalism of general relativity to be
developed inside $E_{8(-24)}$. The model then suggests unexpected and perhaps
subtle ways in which general relativity and particle physics may be forced to
modify each other, in order to produce a unified theory.
