CCFU Proof 28 — Stab(Ω_W) ≅ g₂(split)

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CCFU Proof 28 — Stab(Ω_W) ≅ g₂(split)

05/26/2026

Given.
Ω_W from Proof 22.
dim Stab(Ω_W) = 14 [Proof 23].

G₂ identification (CCFU Script 2):
    1. dim g = 14.
    2. g closed under commutators (Lie subalgebra).
    3. g = [g,g] (perfect).
    4. Killing form nondegenerate (semisimple — Cartan criterion).
    5. R⁷ irreducible (commutant = scalars — Schur).
    6. Simple (ad-commutant dim 1).
    7. Killing signature (8,6) (split real form — Cartan 1914).
    8. g ⊂ so(b_{Ω_W}) (preserves Hitchin metric).

By Killing's classification (1887):
the only 14-dimensional simple Lie algebra (over C) is g₂.
By Cartan's classification of real forms (1914):
Killing signature (8,6) identifies the split real form.

Therefore Stab(Ω_W) has Lie algebra g₂(split).  ∎

External references:
    Killing 1887 (classification of simple Lie algebras over C).
    Cartan 1914 (classification of real forms).
    Cartan criterion (Killing nondegenerate ↔ semisimple).
    Schur's lemma (commutant = scalars ↔ irreducible).

[Dependencies: Proofs 22, 23.]