Here is a non-rigorous, non-technical attempt at an answer:

If CL(n) is the Clifford algebra of an n-dimensional vector space, then Cl(n) is a graded algebra of total dimension 2^n and the grade k part of Cl(n), denoted by Cl(n)k, has dimension n! / k! (n-k)! The Hodge star * map is an automorphism of Cl(n) onto Cl(n) that is an isomorphism between Cl(n)k and Cl(n)(n-k) The grade of the image of the Hodge star * map is called the 8-grade. FOR EXAMPLE: ---------------- 16-DIMENSIONAL Cl(4) has graded dimensions: grade 0 1 2 3 4 dimension 1 4 6 4 1 *-grade 4 3 2 1 0 The Hodge star * map takes the 6-dimensional bivector space into itself. Therefore, the 6-dimensional bivector space splits into two 3-dimensional parts that are interchanged by the Hodge star * map. If the 4-dimensional vector space of Cl(4) is Minkowski space, the two 3-dimensional bivector spaces are the Lie algebras of rotations and boosts. If the 4-dimensional vector space of Cl(4) is Euclidean space, the two 3-dimensional bivector spaces are two copies of SU(2) = Spin(3). Let F be a bivector form. So is *F F is self-dual if F = *F and F is anti-self-dual if F = -*F Let mn be lower indices and MN be upper indices for F. Then *Fmn = (1/2) e(mnab) FAB The INTEGRAL of the trace of F /\ *F over the vector space of Cl(4) is the (negative of) the Yang-Mills action for a pure gauge SU(2) gauge field theory over the vector space of Cl(4). If the vector space of Cl(4) is S4, every self-dual connection of index 1 reduces to the connection SU(2) = Spin(3). An SU(2) = Spin(3) bivector 2-vector space acts as a transitive transformation group of the symmetric space Spin(3) / Spin(2) = S2 and S2 x S2 is a 4-dimensional space with quaternionic structure. ---------------- 256-DIMENSIONAL Cl(8) has graded dimensions: grade 0 1 2 3 4 5 6 7 8 dimension 1 8 28 56 70 56 28 8 1 *-grade 8 7 6 5 4 3 2 1 0 The Hodge star * map takes the 70-dimensional 4-vector space into itself. Therefore, the 70-dimensional 4-vector space splits into two 35-dimensional parts that are interchanged by the Hodge star * map. 256-dimensional Cl(8) can be represented by a 16x16 real matrix algebra. The numbers refer to the grade in Cl(8) of the matrix entry. 0 2 2 2 2 2 2 2 7 5 5 5 5 5 5 5 4 4 2 2 2 2 2 2 5 7 5 5 5 5 5 5 4 4 4 2 2 2 2 2 5 5 7 5 5 5 5 5 4 4 4 4 2 2 2 2 5 5 5 7 5 5 5 5 4 4 4 4 4 2 2 2 5 5 5 5 7 5 5 5 4 4 4 4 4 4 2 2 5 5 5 5 5 7 5 5 4 4 4 4 4 4 4 2 5 5 5 5 5 5 7 5 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 7 * 1 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 3 1 3 3 3 3 3 3 6 4 4 4 4 4 4 4 3 3 1 3 3 3 3 3 6 6 4 4 4 4 4 4 3 3 3 1 3 3 3 3 6 6 6 4 4 4 4 4 3 3 3 3 1 3 3 3 6 6 6 6 4 4 4 4 3 3 3 3 3 1 3 3 6 6 6 6 6 4 4 4 3 3 3 3 3 3 1 3 6 6 6 6 6 6 4 4 3 3 3 3 3 3 3 1 6 6 6 6 6 6 6 8Important Notation Notes: The two blocks of the form 0 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 4 4 4 2 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 2 2 2 4 4 4 4 4 4 2 2 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 are more symbolic than literal. They mean that: the 28 entries labelled 2 correspond to the antisymmetric part of an 8x8 matrix; the 35 entries labelled 4 correspond to the traceless symmetric part of an 8x8 matrix; and the 1 entry labelled 0 corresponds to the trace of an 8x8 matrix. A more literal, but more complicated, representation of the graded structure of those two blocks is: 0 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 2,4 4 However, in the more literal representation, the entries are not all independent. The more symbolic representation is a more accurate reflection of the number of independent entries of each grade. The two blocks of the form 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 1 can be taken more literally, as they mean that: the 8 entries labelled 1 correspond to the diagonal part of an 8x8 matrix; and the 56 entries labelled 3 correspond to the off-diagonal part of an 8x8 matrix.The conventions of the above Notation Notes are used from time to time in my papers and web pages.The Hodge star * map acts like central symmetry of the Cl(8) 16x16 matrix. The spinor space of Cl(8) is a 1x16 column vector. It reduces to two mirror image 1x8 column vectors, the +half-spinor space and the -half-spinor space. + + + + + + + + - - - - - - - - The +half-spinor space is acted on by the elements of Cl(8) of grade 0 1 2 3 4 dimension 1 8 28 56 35 while the -half-spinor space is acted on by the elements of Cl(8) of grade 4 5 6 7 8 dimension 35 56 28 8 1 Since the Hodge star * map interchanges the two sets of elements of Cl(8), the Hodge star * map interchanges the +half-spinor space and the -half-spinor space. That is why the two half-spinor spaces are mirror images of each other. Let F be a bivector form. *F is a 6-vector form. F is self-dual if F /\ F = *(F /\ F) and F is anti-self-dual if F /\ F = -*(F /\ F) Let mn be lower indices and MN be upper indices for F. Then *Fmn = (1/2) e(mnabwxyz) FABWXYZ The INTEGRAL of the trace of F /\ *F over the vector space of Cl(8) is the (negative of) the action for a pure gauge Spin(8) gauge field theory over the vector space of Cl(8). A Spin(8) bivector 2-vector space acts as a transitive transformation group of the symmetric space Spin(8) / Spin(7) = S7 and S7 x RP1 is an 8-dimensional space with octonionic structure. ---------------- D4-D5-E6 MODEL AFTER DIMENSIONAL REDUCTION TO 4-DIMENSIONAL SPACETIME: Dimensional reduction of vector spacetime from 8 to 4 dimensions is done in the D4-D5-E6 model by fixing an associative 3-form and a coassociative 4-form. Since the Hodge star * map takes 3-forms into 5-forms, dimensional reduction removes from the Lagrangian any terms involving Cl(8) elements of grade 3 4 5 dimension 56 70 56 leaving only terms of grade 0 1 2 6 7 8 dimension 1 8 28 28 8 1 Also, since the space spanned by the coassociative 4-form is reduced from spacetime (It forms an internal symmetry space for the gauge groups) the grade of 6-, 7-, and 8-vectors are reduced by 4, the dimension of 1-vectors is reduced to 4, and the dimension of 3-vectors (formerly 7-vectors) is reduced to 4. The resulting structure is grade 0 1 2 3 4 dimension 1 4 56 4 1 NOW, IN THE RESULTING STRUCTURE, THE HODGE STAR * MAP IS DERIVED FROM THE Cl(8) HODGE STAR MAP. Let F be a bivector form. *F is a 2-vector form. F is self-dual if F = *F and F is anti-self-dual if F = -*F Let mn be lower indices and MN be upper indices for F. Then *Fmn = (1/2) e(mnab) FAB The INTEGRAL of the trace of F /\ *F over the 4-dim vector space is the (negative of) the action for a pure gauge Spin(8) gauge field theory over the 4-dim vector space. However, a Spin(8) bivector 2-vector space is too big to act as a transitive transformation group of a symmetric space of the form Spin(8) / G = M where the dimension of M is 4 or less. (Maximal subgroup of Spin(8) is Spin(7).) An SU(3) subgroup of Spin(8) acts as a transitive transformation group of the symmetric space SU(3) / S(U(2)xU(1)) = CP2 and CP2 is a 4-dimensional space with quaternionic structure. An SU(2) subgroup of Spin(8) acts as a transitive transformation group of the symmetric space SU(2) / U(1) = S2 and S2 x S2 is a 4-dimensional space with quaternionic structure. A U(1) subgroup of Spin(8) acts as a transitive transformation group of the symmetric space U(1) = S1 and S1 x S1 x S1 x S1 = T4 is a 4-dimensional space with quaternionic structure. A U(4) subgroup of Spin(8) has 12-dimensional rank-2 coset space Spin(8) / U(4) = M12 M12 corresponds to SU(3) x SU(2) x U(1). U(4) = Spin(6) x U(1) has subgroup Spin(6). Spin(6) acts as the conformal group over the 4-dimensional space RP1 x S3 that is the Shilov boundary of the bounded complex homogeneous domain corresponding to the Hermitian symmetric space Spin(6) / Spin(4) x U(1). A Spin(5) subgroup of Spin(6) acts as a transitive transformation group of the symmetric space Spin(5) / Spin(4) = S4 and S4 is a 4-dimensional space with quaternionic structure. The 5-dimensional coset space Spin(6) / Spin(5) represents the scale and conformal degrees of freedom of the Higgs mechanism. Spin(5) produces gravity by the MacDowell-Mansouri mechanism. If the vector space is S4, every self-dual connection of index 2 is contained in the connection Spin(8). Spin(8) contains BOTH Spin(5) gravity that acts on 4-dim associative spacetime AND SU(3) x SU(2) x U(1) that acts on 4-dim coassociative internal symmetry space. ---------------- If the vector space is S4, every self-dual connection of index 3 is contained in the connection E8. E8 contains the global structure of the 3-fermion-generation D4-D5-E6 model. ----------------

References: Atiyah, Hitchin, and Singer, Self-Duality in Four-Dimensional Riemannian Geometry, Proc. R. Soc. Lond. A362 (1978) 425-461. Gockeler and Schucker, Differential Geometry, Gauge Theories, and Gravity, Cambridge (1987) Grossman, Kephart, and Stasheff, Solutions to Yang-Mills Field Equations in Eight Dimensions and the Last Hopf Map, Commun. Math. Phys. 96 (1984) 4531-437 Nash and Sen, Topology and Geometry for Physicists, Academic Press (1983) Thirring (translated by Harrell) Classical Dynamical Systems and Classical Field Theory vols. 1 and 2 of A Course in Mathematical Physics (2nd edition) Springer-Verlag (1992)

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