Spacetime and Internal Symmetry from Split Bioctonions and the Two Extra SU(3)’s of E₈ × <em>ω</em>E₈

Over the last few years, we have attempted to develop an \( E_8 \times E_8 \) 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 \( E_8\supset E_6\times SU(3) \) on each side of \( E_8\times \omega E_8 \) 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 \( \mathbb{C}P^2=SU(3)/S(U(2)\times U(1)) \). The key algebraic ingredient is the octonionic split \( O=H\oplus H\varepsilon \) with \( \varepsilon\perp H \), by which the branch AdjSU(3) →\( \mathbf{3}_0\oplus \mathbf{2}_{+1}\oplus\overline{\mathbf{2}}_{-1}\oplus \mathbf{1}_0 \) is realised as ℑ\( H\oplus (H\varepsilon)_{\mathbb{R}}\oplus R \). The two \( U(1) \) factors play the role of Spin\( ^c \) connections on the \( \mathbb{C}P^2 \) fibres.