Spacetime and Internal Symmetry from Split Bioctonions and the Two Extra SU(3)’s of E₈ × ωE₈

Over the last few years, we have attempted to develop an \( E_8 × E_8 \) theory of unification to combine the standard model with general relativity. In the present new work, we give a self-contained construction in which the two extra \( SU(3) \) factors that appear in the maximal subgroup chain \( E_8⊃ E_6× SU(3) \) on each side of \( E_8× ω E_8 \) generate: (i) a six-dimensional base \( (M_6,g) \) of signature \( (3,3) \); (ii) two embedded Lorentzian 4D spacetimes; and (iii) per side, a canonical real 4-dimensional internal fibre naturally identified with the tangent of \( CP^2=SU(3)/S(U(2)× U(1)) \). The key algebraic ingredient is the octonionic split \( O=H⊕ Hε \) with \( ε⊥ H \), by which the branch AdjSU(3) →\( 3_0⊕ 2₊₁⊕2̄₋₁⊕ 1_0 \) is realised as ℑ\( H⊕ (Hε)R⊕ R \). The two \( U(1) \) factors play the role of Spin\( ^c \) connections on the \( CP^2 \) fibres. The role of AI in this work is explicitly acknowledged, and highlighted in an appendix.