Arrow: $G_2 \to SO(2,4)$

What is there

This arrow is established in the literature via the split octonions. Gogberashvili (arXiv:1602.07979) shows explicitly that the non-compact (split) form of $G_2$ — the automorphism group of the split octonions — acts on the 8-dimensional split-octonionic space, and that restricting this action to the 7-dimensional imaginary part and projecting to a $(2+4)$-cone generates exactly the 15-parameter conformal group $SO(2,4)$.

The mechanism:

The octonionic 8D interval writes in 6D form as a $(2+4)$-cone whose linear rotations generate all of Poincaré, dilatations, and inversions — the full 15-parameter $SO(2,4)$.

The cosmological constant bonus

The dimensional constant $L$ needed to write the cone form naturally gives the observed value of the cosmological constant. This directly connects to the cosmological redshift / off-axis drift interpretation in epistemics.md.

The compact → split transition: $u$-selection invariant

Both $\mathbb{O}$ (compact) and $\tilde{\mathbb{O}}$ (split) have isomorphic complexification $\mathbb{O}_\mathbb{C}$, since over $\mathbb{C}$ nondegenerate quadratic forms are equivalent (established: Hurwitz/Jacobson). Compact and split $G_2$ are therefore two real forms of the same complex group $G_2(\mathbb{C})$, sharing the same root system and Dynkin diagram.

The working claim is that the $u$-selection operates at the level of $G_2(\mathbb{C})$: it selects a preferred complexified direction and can then be read in both real forms. In that reading, $SU(3)$ is the subgroup of compact $G_2$ that preserves $u$, while $SU(2,1)$ is the stabilizer of the corresponding direction in split $G_2$. These are themselves real forms of the same complex group $SL(3,\mathbb{C})$. This is a structural identification, not an independent proof that one and the same real vector is canonically inherited by both forms.

The two real forms are therefore not two steps in a sequence but two faces of one complex structure, both determined by a single $u$:

G₂(C)  [u selected once at complex level]
     ↙                          ↘
compact G₂                  split G₂
Aut(O)                      Aut(Õ)
internal symmetries         spacetime symmetries
SU(3) color                 SO(2,4) conformal
Jordan / Leech              AdS / holography

Physical reading: The compact face gives internal symmetry structure. The split face gives spacetime symmetry structure. The intended unifier is the same complex-level $u$-selection, though that identification should be read as a structural proposal rather than a separately proved theorem.

The missing duality link

The duality type changes across this arrow: from algebraic duality ($G_2$ ↔ $\mathbb{O}$, group ↔ algebra it acts on) to geometric duality ($SO(2,4)$ ↔ $AdS_5$ boundary, bulk ↔ boundary). The most natural candidate bridging object is the octonion projective plane $\mathbb{O}P^2$, whose isometry group is $F_4$ and whose collineation group is $E_6$. See why-e8.md for the full duality map.

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