Spacetime and Internal Symmetry from Split Bioctonions and the Two Extra SU(3)’s of E8 × ωE8
Abstract
We study a proposed $E_8\times E_8$ kinematic scaffold relating a six-dimensional base to exceptional internal data. The notation $E_8\times\omega E_8$ records a split-complex exchange grading of two factors; it is not a product of a group with a number. The two extra SU(3) factors in $E_8\supset E_6\times SU(3)$ are adopted as geometric structure groups. We select $M_6=\Im\ H_L\oplus\omega\Im H_R$ and separately specify a quadratic form of signature $(3,3)$. Its Clifford algebra is $\mathrm{Cl}(3,3)$, distinct from the eight-dimensional split-biquaternion seed $\mathrm{Cl}(0,3)$. The rank-two Jordan algebra $J_2(Hs)$ realises $(M_6,g)$ through its determinant; after choosing $Hs\cong M_2(\mathbb R)$, its matrix gradient and adjugate act between real four-dimensional Weyl modules and factorise the wave operator exactly. Rank-three $J_3(\mathbb O_{\mathbb C})$ supplies the proposed internal, generation and cubic spectral structure through the magic-star decomposition of $\mathfrak e_8^{\mathbb C}$. Per side, the adjoint branching supplies a real rank-four associated vector bundle whose isotropy representation is isomorphic to a tangent space of $\mathbb CP^2$. Promoting this linear fibre to an associated $\mathbb CP^2$-bundle is an additional geometric choice. The octonionic decomposition $O=H\oplus H\varepsilon$ realises the representation split $\mathbf8\to\mathbf3_0\oplus\mathbf2_{+1}\oplus\overline{\mathbf2}_{-1}\oplus\mathbf1_0$. Soldering, localisation, the real-form…
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