Arrow: $E_8 \to E_6$

What is there

$E_6$ is a maximal rank subgroup of $E_8$, specifically appearing in the decomposition $E_8 \supset E_6 \times SU(3)$. This is standard Lie theory. The $SU(3)$ factor appearing here is already suggestive of color.

Steel man

$E_8$ is the largest exceptional group and the only one self-dual under its adjoint representation. If a single ambient structure is to contain all exceptional symmetry, $E_8$ is the unique candidate. The reduction to $E_6$ is then not arbitrary — $E_6$ is singled out as the exceptional subgroup that naturally houses the Jordan structure, with the $SU(3)$ factor as residue.

Why E₈ rather than E₆ directly

Previously flagged as an unjustified choice. The self-consistency argument (see why-e8.md) closes this gap: E₈ is the unique simple Lie group whose adjoint representation is also its fundamental representation, requiring no external arena. Starting at E₆ would require explaining what E₆ acts on — immediately requiring $J_3(\mathbb{O})$ as an external object. Starting at E₈ requires nothing external. The $SU(3)$ factor in $E_8 \supset E_6 \times SU(3)$ is a residue of the first reduction, already suggestive of color.

Note on the two SU(3)’s: The $SU(3)$ appearing here as the residue of $E_8 \supset E_6 \times SU(3)$ is a different object from the $SU(3) = \mathrm{Stab}_{G_2}(u)$ that appears later in the chain. The former lives outside $E_6$; the latter lives inside $G_2 \subset F_4 \subset E_6$. Whether these two $SU(3)$’s are related — or whether one is the image of the other under the chain — is an open question.

Remaining gap

The self-consistency argument is philosophical rather than a proof. A derivation showing that E₈ self-consistency forces the $u$ selection — rather than merely permitting it — is still missing.

Status