Dixon-Rosenfeld lines and the Standard Model

We present three new coset manifolds named Dixon-Rosenfeld lines that are similar to Rosenfeld projective lines except over the Dixon algebra C⊗H⊗O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}\otimes \mathbb {H}\otimes \mathbb {O}$$\end{document}. Three different Lie groups are found as isometry groups of these coset manifolds using Tits’ formula. We demonstrate how Standard Model interactions with the Dixon algebra in recent work from Furey and Hughes can be uplifted to tensor products of division algebras and Jordan algebras for a single generation of fermions. The Freudenthal–Tits construction clarifies how the three Dixon-Rosenfeld projective lines are contained within C⊗H⊗J2(O)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}\otimes \mathbb {H}\otimes J_{2}(\mathbb {O})$$\end{document}, O⊗J2(C⊗H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {O}\otimes J_{2}(\mathbb {C}\otimes \mathbb {H})$$\end{document}, and C⊗O⊗J2(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}\otimes \mathbb {O}\otimes J_{2}(\mathbb {H})$$\end{document}.


Introduction and Motivation
We focus on the definition of three coset manifolds of dimension 64 that we call Dixon-Rosenfeld lines.Each contains an isometry group whose Lie algebra is obtained from Tits' magic formula.These three constructions are obtained similarly to how projective lines are obtained over R, C, H and O; therefore, they can be thought of as "generalized" projective lines over the Dixon algebra T ≡ C ⊗ H ⊗ O in the sense presented by Rosenfeld in [68][69][70].
Our work on Dixon-Rosenfeld lines defines three homogeneous spaces that locally embed a representation of T to encode one generation of fermions in the Standard Model.Section (2) shows that three coset manifolds of real dimension 64 are possible, giving three non-simple Lie algebras as isometry groups that are obtained from Tits formula.Section (3) analyzes the relationship between the new Dixon-Rosenfeld lines with the Rosenfeld lines.Section (4) uplifts scalar, spinor, vector, and 2-form representations of the Lorentz group representations with C ⊗ H from Furey [37] to C ⊗ J 2 (O).Section (5) uplifts the Standard Model fermionic charge sector described by Furey with C ⊗ O [39] to C ⊗ J 2 (O).Section (6) uplifts recent work by Furey and Hughes for encoding Standard Model interactions with C ⊗ H ⊗ O [44] to the three different realizations of the Dixon-Rosenfeld lines via C ⊗ H ⊗ J 2 (O), O ⊗ J 2 (C ⊗ H), and C ⊗ O ⊗ J 2 (H).Section (7) concludes with a summary of our work and outlines prospects for future work.

Tensor products on unital composition algebras
An algebra is a vector space X with a bilinear multiplication.Different properties of the multiplication give rise to numerous kind of algebras.Indeed, for what it will be used in the following sections, an algebra X is said to be commutative if xy = yx for every x, y ∈ X, is associative if satisfies x (yz) = (xy) z, is alternative if x(yx) = (xy)x, flexible if x(yy) = (xy)y and, finally, power-associative if x(xx) = (xx)x and (xx)(xx) = ((xx)x)x.It is worth noting that the last four proprieties are progressive and proper refinements of associativity, i.e. associative ⇒ alternative ⇒ flexible ⇒ power-associative.
Every algebra has a zero element 0 ∈ X, since X has to be a group in respect to the sum, but if it also does not have zero divisors, then X is called a division algebra, i.e. if xy = 0 then or x = 0 or y = 0.While the zero element is always present in any algebra, if it exists an element 1 ∈ X such that 1x = x1 = x for all x ∈ X then the algebra is unital.Finally, if we can define over X an involution, called conjugation, and a quadratic form N , called norm, such that with x, y ∈ X and x as the conjugate of x, then the algebra is called a composition algebra.
A well-known theorem due to Hurwitz [51] states that R, C, H and O are the only four normed division algebras that are also unital and composition [7,33].More specifically, R is also totally ordered, commutative and associative, C is just commutative and associative, H is only associative and, finally, O is only alternative, as summarized in Table (1).
Since all four normed division algebras are vector spaces over the field of reals R we are able to define a tensor product A ⊗ B of two normed division algebras, with a bilinear product defined by where a, c ∈ A and b, d ∈ B. Every alternative algebra tensor a commutative algebra yields again to an alternative algebra, so that with few additional efforts we can easily find all properties for triple tensor products listed in Table (2).
It is straightforward to see that every element in the set D = {(I q ± 1) , (I e α ± 1) , (qe α ± 1) : q ∈ {i, j, k}} , is a zero divisor and therefore T is not a division algebra.Moreover, the Dixon algebra is not commutative, neither associative, nor alternative or flexible and, finally, not even power-associative, i.e. in general x (xx) ̸ = (xx) x .Nevertheless, it is possible to define a quadratic norm N over T, starting from the decomposition in Eq. ( 5), i.e.
with an associated polar form ⟨•, •⟩ given by the symmetric bilinear form 2 Dixon-Rosenfeld lines The geometrical motivation for defining Dixon-Rosenfeld lines as coset manifolds relies on the study of the octonionic planes explored by Tits, Freudenthal and Rosenfeld in a series of seminal works [36,68,75,79] that led to a geometric interpretation of Lie algebras and to the construction of the Tits-Freudenthal Magic Square.While Freudenthal interpreted the entries of the Magic Square as different forms of automorphisms of the projective plane such as isometries, collineations, homography etc., Rosenfeld thought of every row of the magic square as the Lie algebra of the isometry groups of a "generalized" projective plane over a tensorial product of Hurwitz algebras [68] (see also [60] for a recent systematic review).In fact, tensor products over Hurwitz algebras are not division algebras, which therefore do not allow the definition of a projective plane in a strict sense.Nevertheless, later works of Atsuyama proved the insight of Rosenfeld to be correct and that it is possible to use these algebras to define projective planes in a "wider sense" [1,3,57].A similar analysis was then carried out for generalized projective lines making use the Tits-Freudenthal Magic Square of order two instead of three, thus relating the resulting Lie algebras with isometries of generalized projective lines, instead of planes (see [60], for more details).

Dixon lines as coset manifold
Coset manifolds arise from coset spaces over a Lie group G given by an equivalence relation of the type where g, g ′ ∈ G and h ∈ H and H is a closed subgroup of G.In this case, the coset space G/H, obtained from the equivalence classes gH, inherits a manifold structure from G and is therefore a manifold of dimension Moreover, G/H can be endowed with invariant metrics such that all elements of the original group G are isometries of the constructed metric [34,60].More specifically, the structure constants of the Lie algebra g of the Lie group G define completely the metric and therefore all the metric-dependent tensors, such as the curvature tensor, the Ricci tensor, etc.Finally, the coset space G/H is a homogeneous manifold by construction, i.e. the group G acts transitively, and its isotropy subgroup is precisely H, i.e. the group H is such that for any given point p in the manifold hp = p.Therefore, for our purposes in the definition of the Dixon-Rosenfeld lines, it will be sufficient to define the isometry group and the isotropy group of the coset manifold to have them completely defined in its topological and metrical descriptions.

Tits' magic formula
We now proceed defining three Dixon projective lines as three different coset spaces of real dimension 64 obtained from three isometry algebras a I , a II and a III making the use of Tits' magic formula [75] for n = 2, i.e.
where A, B are alternative algebras and J 2 (B) is a Jordan algebra over Hermitian two by two matrices2 .Brackets on L 2 (A, B) can be defined following notation in [15, sec. 3] for which, given the an algebra A, we define as the projection of an element of the algebra in the subspace orthogonal to the identity denoted as 1.We then define J ′ 2 (B) the algebra obtained by such elements with the product • given by the projection back on the subspace orthogonal to the identity of the Jordan product, i.e.
where as usual, we intended . With this notation, the vector space ( 14) is endowed with the following brackets [15] 1.The usual brackets on the Lie subalgebra der (A) ⊕ der (J 2 (B)).2. When a ∈ der (A) ⊕ der (J 2 (B)) and

When
where L x and R x are the left and right action on the algebra and D x,y is given by Applying now formula (19) to the Jordan algebra with left and right Jordan product, the left and right products are the same and and 3 Isometry and isotropy Lie algebras of the three Dixon-Rosenfeld lines.For TP 1 II , the "minimal" enhancement (51) is considered.

Three isometry groups
Tits' formula is the most general formula compared to those of Vinberg [79], Atsuyama [2], Santander and Herranz [73], Barton and Sudbery [15], and Elduque [31] since it does not require the use of two composition algebras, but only the use of an alternative algebra and a Jordan algebra obtained from another alternative algebra.
Next, we consider all tensor products of the form A⊗J 2 (B) with A and B alternative such that A ⊗ B corresponds to the Dixon algebra C ⊗ H ⊗ O. Since H ⊗ O is not alternative, the possible candidates can be a priori only related with the following four different A and B, i.e.
and, finally, A = H, B = (C ⊗ O) .However the latter case, i.e.A = H, B = (C ⊗ O), would need the existence of a Jordan algebra J 2 (C ⊗ O) over bioctonions C⊗O, which is not possible. 3e are therefore left with only three different possibilities, i.e.
We will now discuss how, due to the three possible cases in Eq. ( 25), there exist three "homogeneous realizations" of the Dixon-Rosenfeld projective line TP 1 , which will be distinguished by the subscript I, II and III, respectively.
We start and observe that implying that the imaginary biquaternions are This can be understood by observing that Note that der (C ⊗ H) is next-to-maximal into der (O) ≃ g 2 , because it can be obtained by a chain of two maximal (and symmetric) embeddings, or equivalently by a chain of two maximal (one non-symmetric and one symmetric) embeddings, In all cases, the Dixon algebra T will have the same covariant realization in terms of der (T) ≃ der i.e.T ≃T TP 1 which can enjoy the following enhancements of (manifest) covariance, I. In the case A = C ⊗ H and B = O, Tits' formula ( 14) yields (cf.(26)) because The Lie algebra isom TP 1 I has therefore dimension 3 + 36 + 63 = 102.II.In the case A = O and B = C ⊗ H, after the treatment given in Sec. 8 of [15], Tits' formula (14) gets der (O) replaced by so (O ′ ), and thus one obtains J 2 (C ⊗ H) is a rank-2 Jordan algebra, defined as the algebra of 2 × 2 matrices over C ⊗ H (cf. ( 27) and ( 29)) which are Hermitian with respect to the involution ı given by the composition of the conjugation of C and of the conjugation of H: Interestingly, this implies that the diagonal elements of the matrices of J yielding for the traceless part that On the other hand, as proved in Appendix B, it holds that and thus (41) and (42) respectively enjoy the following enhancements5 : Therefore, since O ′ ≃ 7 of so 7 = 7 of g 2 , formula (39) can be made explicit as follows: = g 2 ⊕ so 6 ⊕ 3 • (7, 4) ⊕ (7, 1) The last line (49) has a manifest (g 2 ⊕ su 2 )-covariance, which is the natural one for the Dixon algebra T (cf.(33)), giving because there is only one singlet (1, 1) in (49).See Appendix C for a more exhaustive treatment of all su 2 's inside so 6 .However, it is anticipated that T ∈ a II .Therefore, there are two possibilities to resolve this issue.First, it was asserted above that H corresponds to 1⊕3 of su 2 , which led to T ∈ a I .However, C ⊗ H is known to allow for three different representations of sl 2,C [37].If the spinor representations were chosen instead, then T ∈ a II .Second, one may claim that the 2 × 2 Freudenthal-Tits formula does not apply to the case where A = O and B = C ⊗ H. Freudenthal and Tits' formula was designed for 3 × 3, but the 2 × 2 case already has a precedent of the formula depending on the algebras chosen, as A = O leads to a difference from A = C or H in the 2 × 2 case.In this work, we merely claim that a non-simple Lie algebra a II exists, but we do not fully determine its precise structure.
Note however that a further symmetry enhancement to so 8 ⊕ su 2 is not possible without breaking T itself 6 .If the Lie algebra a II should be enhanced, then a II,enh.≡ isom TP 1 II enh.given by ( 52) has dimension 21 + 15 + 3 • 28 + 1 = 121.Alternatively, it is possible that a II is 120-dimensional such that H ∈ T contains spinor representations, such as 2 ⊕ 2 of der(H) = su 2 .III.Finally, in the case A = C ⊗ O (non-associative) and B = H, Tits' formula (14) yields because and The Lie algebra isom TP 1 III has dimension 14 + 10 + 75 = 99.

Three Dixon lines
A Dixon-Rosenfeld projective line TP 1 can be realized as an homogeneous space of dimension dim R TP 1 = dim R T = 64, whose corresponding Lie algebra generators Lie TP 1 relate to the isometry and isotropy Lie algebras as follows: and whose tangent space T TP 1 carries a isot TP 1 -covariant realization of T itself.
I By iterated branchings of isom TP 1 I given by (36), one obtains thus implying that isot TP 1 c TP 1 Therefore, one obtains the following (non-symmetric) presentation of the Dixon projective line TP 1 I as a homogeneous space: with dim TP 1 The coset ( 63) is not symmetric, because it can be checked that where subscript "a" denotes anti-symmetrization of the tensor product throughout.II From ( 48) and ( 52), the "minimally" enhanced isom TP 1 with c TP 1 Thence, one obtains the following (non-symmetric) presentation of the Dixon projective line TP 1 II as a homogeneous space: once again with dim TP 1 The coset ( 69) is not symmetric, because it can be checked that III By iterated branchings of isom TP 1 III , given by ( 54), one obtains thus implying that c TP 1 and therefore leading to the following (non-symmetric) presentation of the Dixon projective line TP 1 III as a homogeneous space: once again with dim TP 1 The coset ( 75) is not symmetric, because it can be checked that Remark The above analysis yields the following isometry algebras: isom TP 1 isom TP 1 isom TP 1 as well as the following isotropy algebras: isot TP 1 isot TP 1 isot TP 1 which all imply the same coset Lie algebra locally on the tangent space, providing a manifestly (g 2 ⊕ su 2 )-covariant (or, equivalently, (so 7 ⊕ su 2 )-covariant) realization of the Dixon algebra T, as given by ( 33) and ( 34): Thus, the three Dixon-Rosenfeld projective lines TP 1 I , TP 1 II and TP 1 III have slightly different isometry and isotropy Lie algebras; from the formulae above, it follows that isom TP 1 and isot TP 1 However, the set of generators of the isometry Lie group whose non-linear realization gives rise to the Dixon-Rosenfeld projective line is the same for TP 1 I , TP 1 II and TP 1 III ; since such a set of generators also provide a local realization of the tangent space, one can conclude that TP 1 I , TP 1 II and TP 1 III are locally isomorphic as homogeneous (non-symmetric) spaces.

Relationship with octonionic Rosenfeld lines
It is interesting to point out the relationship between the Dixon-Rosenfeld lines and the other octonionic Rosenfeld lines, whose definition can be found in from an historical point of view in [69,70] and in a more rigorous definition in [60].Let us just recall here the homogeneous space realization of Rosenfeld lines over A ⊗ O, with A = R, C, H, O (see [69,70] and [23,24,60]), i.e. for the octonionic projective line (R ⊗ O) P 1 , the bioctonionic Rosenfeld line (C ⊗ O) P 1 , the quateroctonionic Rosenfeld line(H ⊗ O) P 1 and, finally, for the octooctonionic Rosenfeld line from which it consistently follows that (95) which illustrates how the tangent spaces of octonionic projective lines generally carry an enhancement of the symmetry with respect to the Lie algebra der Geometrically, the octonionic projective lines (A ⊗ O) P 1 can be regarded as A⊗O together with a point at infinity, and thus as a 8dim R A-sphere, namely as a maximal totally geodesic sphere in the corresponding octonionic Rosenfeld projective plane (A ⊗ O) P 2 [70].In the case A = R, such a "spherical characterization" of octonionic projective lines is well known, whereas for the other cases (the "genuinely Rosenfeld" ones) it is less trivial (see e.g.[65]).
We can now study the relations among the Dixon-Rosenfeld lines discussed above and the octonionic Rosenfeld lines.Of course, By recalling (33) and considering the ( one observes that, when restricting the first (or, equivalently, the second) g 2 to a su 2 subalgebra defined by (30) (or, equivalently, by ( 31)), the irrepr.7 of g 2 breaks into 2 • 3 + 1 of su 2 , and therefore it holds that ≃ T.
In other words, as resulting from the treatment below, O ⊗ O and T are isomorphic as vector spaces (but not as algebras), with der (O ⊗ O) ⊋ der (T): thus, octo-octonions have a larger derivation algebra than the Dixon algebra, with an enhancement/restriction expressed by (30) or, equivalently, by (31).
Thus, it holds that isom TP 1 isot TP 1 and II Analogously, from ( 53) and ( 67), one respectively obtains and However, III Again, from ( 54) and ( 73), one respectively obtains and isot and In other words, the Dixon-Rosenfeld projective lines TP 1 I and TP 1 III have the isometry resp.isotropy Lie algebra strictly contained in the isometry resp.isotropy Lie algebra of the octo-octonionic projective line (O ⊗ O) P 1 , whereas the Dixon-Rosenfeld projective line TP 1 II does not contain nor is contained into (O ⊗ O) P 1 .Nonetheless, as pointed out above, the set of generators of the isometry Lie group whose nonlinear realization gives rise to the Dixon-Rosenfeld projective line is the same for TP 1 I , TP 1  II and TP 1 III ; thus, one can conclude that all such spaces are locally isomorphic as homogeneous spaces: It is interesting to remark that this holds notwithstanding the fact that, while the three Dixon-Rosenfeld projective lines have non-symmetric presentations, the octooctonionic Rosenfeld projective line is a symmetric space.[37].Given some algebra g, a (generalized) minimal ideal i ⊂ g is a subalgebra where m(a, v) ∈ i for all a ∈ g and v ∈ i with m as a (generalized) multiplication.The generalized minimal left ideal that Furey considered for spinors from where P = (1 + Ik)/2 such that P * = (1 − Ik)/2 are projectors satisfying P 2 = P, P * 2 = P * , and P P * = P * P = 0.The 4-vectors (1-forms) were found as generalized minimal ideals via the the following generalized multiplication, where a † = â * is used just for this subsection when a ∈ C⊗H, withˆand * denoting the quaternionic and complex conjugate, respectively.The symbol † is used throughout as a Hermitian conjugate of the algebra, but the explicit mathematical operation will differ depending on the algebra under consideration.The scalars and field strength (2-forms) were found as generalized minimal ideals via the generalized multiplication below, Focusing on the spinors, a Dirac spinor ψ D as an element of C ⊗ H is decomposed into left-and right-chiral (Weyl) spinors ψ L and ψ R as minimal left ideals with respect to Eq. ( 115), where c i for i = 1, . . .
Additionally, the Lorentz transformations can be found as the exponentiation of linear combinations of vectors and bivectors of Cl(3).
The basis of minimal ideals is less clear with C ⊗ H and improved with reference to another basis spanned by {P, P * , jP, ȷP * , IP, IP * , IjP, I ȷP * }.To provide a dictionary of various representations used by Furey for the spinor minimal ideal bases [37][38][39], consider We found it convenient to confirm that ψ L and ψ R are minimal left ideals in Mathematica when converting to the basis above (along with the four elements multiplied by I ).The following anti-commutation relations can be found, α † , α † = 0.
Note that Ii and Ij act as bases of Cl(2).
Note that here † denotes matrix transpose and quaternionic conjugation.This brings in a complication for generalizing P , as 2 × 2 matrices admit two projectors as idempotents, yet C ⊗ J 2 (H) does not contain P = (1 + Ik)/2 on any diagonal elements.The action of C ⊗ H must occur on the off-diagonals.Despite not giving projectors, the bases are embedded as follows A new generalized multiplication was identified for spinors as elements of C⊗J 2 (H) by taking the Jordan product with two matrices from the right to replace P and P * in Eq. (115), where a ∈ C ⊗ J 2 (H) and a • b = (ab + ba)/2 is the Jordan product.We verified in Mathematica that m 4 (a, v) gives spinorial ideals for arbitrary a ∈ C⊗J 2 (H).Since C⊗ J 2 (H) is larger than the piece of C ⊗ H embedded in C ⊗ J 2 (H), the existence of such a generalized ideal may hold for the entire algebra constructed from the Dixon-Rosenfeld line via the Freudenthal-Tits construction.
For Hermitian and anti-Hermitian vectors, the following generalized multiplication rule is found, where m 5 is identified as a Jordan anti-associator.If a is chosen to be a purely offdiagonal element of C ⊗ J 2 (H), then m 5 leads to an element of i for v as a Hermitian or anti-Hermitian vector.If a is chosen as an arbitrary element of C ⊗ J 2 (H), then the Hermitian vector uplifted to C ⊗ J 2 (H) develops a purely real diagonal term, while the antiHermitian vector uplifted develops a purely imaginary diagonal term.It is also anticipated that diagonals of C ⊗ J 2 (H) not found in C ⊗ H should be purely bosonic, which motivates a higher-dimensional Hermitian and anti-Hermitian vector to be found as ideals of C ⊗ J 2 (H).
For scalars and two-forms, the following generalized multiplication rule is found with a Jordan anti-associator and slightly different conjugation, It turns out that the 2-form uplifted to C ⊗ J 2 (H) is a minimal ideal, while the scalar uplifted must be generalized to include a complex diagonal.
For concreteness, the left-and right-chiral spinors embedded in C ⊗ J 2 (H) as minimal ideals of m 4 in Eq. ( 123) are The vectors h µ and pseudo-vectors g µ for µ = 0, 1, 2, 3 represented as elements of C ⊗ H to be used with Eq. ( 124) are generalized to the following minimal ideals of C ⊗ J 2 (H) with diagonal components where h 4 , h 5 , g 4 , and g 5 are scalar degrees of freedom found on the diagonals of the minimal ideals that extend the 4-vector and 4-pseudo-vector.The scalars ϕ and 2forms F embedded in C ⊗ J 2 (H) with Eq. (125) are found as minimal ideals when a complex diagonal is added to the scalars One may anticipate that the vector, spinor, and conjugate spinor representations can be embedded in the three independent off-diagonal components of C ⊗ J 3 (H), but this is left for future work.To establish our conventions for octonions, we review the complexification of the octonionic chain algebra applied to raising and lowering operators for SU (3) c × U (1) em fermionic charge states [37,39].For C ⊗ O, we use I and e i for i = 1, . . ., 7 as the imaginary units.To convert from Furey's octonionic basis to ours, take {e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } → {e 2 , e 3 , e 6 , e 1 , e 5 , e 7 , −e 4 }.A system of ladder operators was constructed from the complexification of the octonionic chain algebra C ⊗ ← − O ∼ = Cl(6), which allows contact with SU (3) c × U (1) em [40].Due to the nonassociative nature of the octonions, the following association of multiplication is always assumed, where an arbitrary element f ∈ C ⊗ O must be considered, If a * refers to complex conjugation and ã refers to octonionic conjugation, denote a † = ã * as the Hermitian conjugate only when acting on a ∈ C ⊗ O.
Our basis of raising and lowering operators is chosen as With this basis, we explicitly confirmed in Mathematica that the following relations hold, It was also confirmed that {α * i , αj } = δ ij .For later convenience, a leptonic sector of operators is also introduced as Due to the non-associativity of octonions, acting from the left once does not span all of the possible transformations, which motivates nested multiplication.This naturally motivates C ⊗ ← − O as the octonionic chain algebra corresponding to Cl(6).This chooses −e 4 as a pseudoscalar, such that the k-vector decomposition of Cl( 6) is spanned by 1-vectors {Ie 2 , Ie 3 , Ie 6 , Ie 1 , Ie 5 , Ie 7 }.
Next, a nilpotent object ω = α 1 α 2 α 3 is introduced, where the parentheses of the chain algebra mentioned above is assumed below.The Hermitian conjugate is The state v c = ωω † is considered roughly as a vacuum state (perhaps renormalized with weak isospin up), since α i ωω † = 0. Fermionic charge states of isospin up are identified as minimal left ideals via where ν, di , u i , and ē are complex coefficients.The weak isospin down states are found by building off of v * c = ω † ω, giving These algebraic operators represent charge states associated with one generation of the Standard Model with reference to SU (3) c × U (1) em .
A notion of Pauli's exclusion principle is found, since the following relations hold, The above equations imply that it is impossible to create two identical fermionic states.As implied, the three raising/lowering operators are associated with three color charges.Furey also demonstrated that the electric charge is associated with the mean of the number operators N i = α † i α i [40].To obtain spinors associated with these charge configurations, Furey advocates for

Uplift of
Next, the analogous raising and lowering operators associated with one generation of the Standard Model are constructed with elements of C⊗ ←−−− J 2 (O).Our guiding principle is to take elements of C ⊗ O, place them on the upper off-diagonal component of C ⊗ ←−−− J 2 (O), and add the Hermitian octonionic conjugate.We seek a new generalized multiplication that implements the same particle dynamics as C⊗ ← − O .For concreteness, consider J f as an arbitrary element of C ⊗ J 2 (O), where The Jordan product is utilized to restore elements of C ⊗ J 2 (O).However, this conflicts with left multiplication utilized in the chain algebra where J 1 , J 2 ∈ C⊗J 2 (O) as arbitrary elements.Rather than having a single element of C ⊗ J 2 (O) to implement α i and α † j , the multiplication above is utilized.The following C ⊗ O variables are first uplifted to elements of C ⊗ J 2 (O), where α 0 = −e 4 + I was introduced for later convenience.We also introduce J Iαi = IJ αi as a shorthand.
These matrices allow for the following nested multiplications to mimic the action of α i and α † j , The following anticommutation relations were explicitly verified, where J off f contains only the off-diagonal components of J f .This suffices to generalize the fermionic degrees of freedom from C⊗O since they are uplifted to the off-diagonals of C ⊗ J 2 (O).
As an abuse of notation, m αi m αj is shorthand for m αi m αj (J f ) .The nilpotent One may verify that m ω m ω = m ω † m ω † = 0, while m ω m ω † acts on J f to give a where f is the upper-right component of J f and ωω † f * † is a shorthand for the octonionic conjugate.This allows for the assignment of a neutrino "vacuum" state, which allows for the following assignments of particles, and In summary, the collection of weak-isospin up and down states are where ν, dr , etc. are complex coefficients.
6 Projective lines over Furey provided a formulation of the electroweak sector [41], which led to the Standard Model embedded in SU (5) and allows for U (1) B−L symmetry [43,47].The construction relies on identifying Cl(10) = Cl(6) ⊗ C Cl(4), which can be found from a double-sided chain algebra over C ⊗ H ⊗ O.For instance, the left-and rightchiral spinors can be brought together via ψ D = ψ R + ψ L with the gamma matrices implemented as where a|b acting on z is azb, which is well-defined when (az)b = a(zb).This allows for left and right action of C ⊗ H to give Cl(4) = Cl(2) ⊗ C Cl(2).This idea can be taken further to give Cl (10) to identify Spin(10) and make contact with SU (3)×SU ( 2 Recent work by Furey and Hughes introduced fermions in the non-associative C ⊗ H ⊗ O algebra to solve this fermion doubling problem, which can be resolved by taking a slightly different route to Spin (10), rather than taking bivectors of Cl (10) [44].Instead, consider the following generalization of Pauli matrices, where i = 1, . . ., 7 and {σ i , σ 8 , σ 9 } allow for a basis of Cl( 9).The ten "generators" σ I for I = 1, . . ., 10 lead to transformations on f via where σa = −σ a for a = 1, . . ., 9 and σ10 = σ 10 .This allows for Spin (10) to act on a Weyl spinor in the 16 representation instead of two 1-component objects of 16 ⊕ 16 to resolve the fermion doubling problem.
With α µ = (Il * , q 1 , q 2 , q 3 ) and α * µ = (−Il, q * 1 , q * 2 , q * 3 ) for µ = 0, 1, 2, 3 as an electrostrong sector and ϵ αβ with α =↑, ↓ as an electroweak sector, the non-associative algebra C ⊗ H ⊗ O can be used to implement particle states for a single generation of the Standard Model fermions.We specify the particle states by using the notation and assignments recently introduced by Furey and Hughes in their solution to the fermion doubling problem [44], namely The coefficients such as V ↑ L are complex.In our conventions, the SU (3) Gell-Mann matrices are represented as elements of where e ij f stands for e i (e j f ).For the electroweak sector with SU (2)×U (1) symmetry, the SU (2) generators are represented in terms of imaginary quaternions and a weak isospin projector s = (1 − Ie 4 )/2, The weak hypercharge is given by Note that all operators from SU (3 O and act from the left.The electric charge operator Q is By separating ψ into ψ l + ψ q + ψ c ν + ψ c e + ψ c u + ψ c d , the following fields are found to correspond to the appropriate representations of the Standard Model, (1, 1) 1 : where we confirmed that the above states have the appropriate weak hypercharge values as well as weak isospin and electric charges.Note that complex conjugation leads to the appropriate conjugate states, which turns left(right)-chiral particles into right(left)-chiral anti-particles.Finally, the largest algebra commuting with so 10 derived from C ⊗ H ⊗ ← → O when considering action from the left and right is given by sl 2,C , which are generated by {1|i, 1|j, 1|k, 1|Ii, 1|Ij, 1|Ik}.

Uplift to C ⊗ H ⊗ J 2 (O)
To uplift the physics of C⊗H⊗O to C⊗H⊗J 2 (O), we start by considering f ∈ C⊗H⊗O uplifted to an off-diagonal matrix . Our first goal is to understand how to implement left multiplication of C⊗H⊗O basis elements on f by the analogous construction in C ⊗ H ⊗ J 2 (O) acting on J off f , where For C⊗H bases, these can be implemented by mapping the basis elements to the same elements times the identity matrix.The same cannot be done for O, as the elements e i must map to J 2 (O) via the eight off-diagonal octonionic Pauli matrices J ei , To understand how to multiply f from the left by e i generalized to C ⊗ H ⊗ J 2 (O), the Fano plane is crucial.A single octonionic unit can always be implemented by multiplying by two units in four different ways.For instance, e 1 = e 1 1 = e 2 e 3 = e 4 e 5 = e 7 e 6 .If e 1 f is uplifted to J off e1f , by recalling the definition (137) of nested commutator of Jordan products, a generalized multiplication rule can be found to give J off e1f from J off f , Above, J 1 represents the uplift of 1 to the real traceless symmetric 2 × 2 matrix, not an arbitrary element.Even though we are implementing octonionic multiplication, the above relations hold for f ∈ C ⊗ H ⊗ O.This allows for a representation of the Gell-Mann matrices in terms of elements of C where )) more precisely.From here, particle states associated with elements of C ⊗ H ⊗ O can be uplifted to C ⊗ H ⊗ J 2 (O).It was confirmed that the SU (3) generators above annihilate leptons and apply color rotations to the quarks in the appropriate manner.
The same relations found in C ⊗ H ⊗ O for SU (2) × U (1) generators are also found by the appropriate uplift to C ⊗ H ⊗ ←−−− J 2 (O).The appropriate left action of g ∈ C ⊗ H on f uplifted to J off f can be found simply by taking gJ off f , since the diagonal elements of C⊗H⊗J 2 (O) can contain C⊗H.Uplifting the generators of SU (2)×U (1) therefore gives where all multiplication is assumed to act from the left.Similarly, the electric charge operator becomes The fermionic states in the C ⊗ H ⊗ J 2 (O) are identified as (1, 1) 1 : where in our conventions, the C ⊗ O quantities such as l and q a are uplifted explicitly to give It was confirmed that m τ11 , m Y , and m Q give the appropriate eigenvalues for these states. 6.
where f * is the complex conjugate and f is the quaternionic conjugate.Finding the corresponding left action of Left multiplication of I on f uplifted to J off If must be implemented with the nested Jordan commutator product (137), This holds for arbitrary elements f ∈ C ⊗ H ⊗ O.The analogous relationship for imaginary quaternionic units are The corresponding uplift of left multiplication by imaginary octonions is given by left multiplication, such that J off eif = e i J off f .From here, the uplift of the fermionic states and the action of bosonic operators on the fermions is similar to the previous discussion on C ⊗ H ⊗ J 2 (O).To highlight this uplift with more detail and for a specific example, consider ψ c e as a left-chiral positron and weak isospin singlet, The action of the Gell-Mann generators uplifted to The electroweak generators are given by where The electric charge operator is given by The action of these generators leads to the expected results when acting on J ψ c ν .For instance, all of the SU (3) generators vanish and J ψ c ν is an eigenstate of m τ11 and m Y , where 1 is found as an eigenvalue for electric charge and weak hypercharge with the left-chiral positron.

Uplift to
where, as above, f denotes the quaternionic conjugation of f .From here, it is clear that the uplift of left multiplication by imaginary quaternionic units is identical to Eq. ( 165).Less care is needed with the complex numbers and octonions, as they are on the diagonals of C ⊗ O ⊗ J 2 (H).
The action of the Gell-Mann generators uplifted to C ⊗ O ⊗ J 2 (H) is identical to Eq. ( 150).The electroweak generators are given by The electric charge operator is given by The fermions of C ⊗ H ⊗ O can be uplifted to C ⊗ O ⊗ J 2 (H) via Eq.( 173) and the generators shown above can be found to act appropriately on the fermionic states.

Conclusions
In this work, we showed how to construct three homogeneous spaces that, following Rosenfeld's interpretation of the Magic Square, correspond to his "generalized" projective lines over the We provided explicit states for one generation of fermions in the standard model within these projective lines, including operators for gauge boson interactions and identification of charges.
While non-simple Lie algebras were found from the Dixon-Rosenfeld projective lines and one generation of the Standard Model fermions were embedded into these projective lines, further work is needed to see if the appropriate representations of the Standard Model are contained within the corresponding isometry groups.For instance, while the bosonic interactions with fermions were demonstrated to be in the chain algebras over division algebras tensored with Jordan algebras and various SU (3) × SU (2) × U (1) groups can be found in the derivation groups, the representations with respect to these groups do not isolate the Standard Model fermionic representations and charges.This is similar to how Spin(9), SU (3) × SU (3), and F 4 are not GUT groups, but the octonions and F 4 have been used to encode Standard Model fermions [54,77,78].
It appears that the Freudenthal-Tits formula should work for A = O and B = C⊗H to give a Lie algebra a II .However, there is not a single formula for the 2 × 2 case, as setting A = O already leads to a difference.Here, we articulated the structure of J 2 (C ⊗ H) and found der(J 2 (C ⊗ H)).However, applying the 2 × 2 analogue of the Freudenthal-Tits construction did not lead to the anticipated representations of T with respect to der(T).To further complicate matters, it is known that C ⊗ H can lead to multiple representations.For now, we merely claim that some non-simple Lie algebra a II exists that contains at least 120 dimensions.By exploring the 3 × 3 case in future work, we hope to gain a further understanding of the true definition of a II .
Additional work is needed to see if other subalgebras of these non-simple Lie algebras exist that can isolate the appropriate representation theory for the Standard Model.Otherwise, chain algebras such as ← −−−−−− − A ⊗ J 2 (B) may lead to Clifford algebras that would be large enough to contain the Standard Model gauge group, just as C ⊗ H ⊗ ← − O can lead to Cl (10).In future work, we seek to investigate the notion of Dixon-Rosenfeld projective planes to see if this may provide applications for three generations of the Standard Model fermions with C ⊗ H ⊗ O. Interactions with the Higgs boson would also be worth exploring, which has been discussed recently [45].
A J 2 (C ⊗ H) as 4 × 4 complex matrices The Jordan algebra J 2 (C ⊗ H) is 16-dimensional and can be expressed in terms of a set of matrices in M 4 (C).We review this isomorphism and determine the action of the double conjugation with respect to C and H in the language of 4×4 complex matrices.Before introducing J 2 (C ⊗ H), we first clarify how the double conjugation of C ⊗ H with respect to C and H leads to a 4-dimensional element and specify how this maps into the isomorphism with M 2 (C).
First, f ∈ C ⊗ H is recast in M(f ) ∈ M 2 (C) by the following isomorphism, using our notation of ( 29) and (118): Next, we clarify how double conjugation of C ⊗ H maps into M 2 (C), As shown above, f * ∼ = M(f ) ⊤ .By using the conventions of conjugation as shown in Eqs. ( 115)-(116), we find a "real" element of 4 dimensions by where r and s real with respect to the double conjugation on C ⊗ H, leading to r and s as 4-dimensional elements spanning {1, Ii, Ij, Ik} with M As such, we refer to X as X(r, s, f ).
B Demonstration of der (J 2 (C ⊗ H)) ∼ = su (4)   The derivation of an alternative algebra is defined ([71] page 77) as a Bracket algebra satisfying the Leibniz rule, and a theorem shows it is of the form When the Jordan algebra is not only alternative but is commutative, L X = R X and the inner derivations are ( [71] page 92) A derivation parameterized by X and Y applied to an element where [X, Z, Y ] is the Jordan associator, sandwiching Z between X and Y .
The solution for the matrix components above are found by plugging Eq. (184) into Eq.(185) to give where the solutions to the other six ϕ i,j can be found similarly.While there is not a unique set of matrices, we found a small collection of X a for a = 1, . . ., 4 and Y b for b = 1, . . .12 that lead to the 15 generators L A .X a are given by X 1 = X(0, 0, 1) = X(0, 0, f 1,1 = 1), X 2 = X(0, 0, I) = X(0, 0, f 1,2 = 1), The three Cartan generators of su 4 are the three first expressed above, diagonal and commuting.By construction as commutators of Hermitian matrices scaled by the imaginary factor i, the matrices L a,b are all Hermitian, and span at most the 16dimensional space u 4 , but from the property that ρ 1,1 + σ 1,1 = 0 and that M(δ) is traceless, the non-traceless generator of u 4 can not be obtained as a derivation L a,b , and therefore the derivation of J 2 (C ⊗ H) is su 4 .
C Demonstration that the algebra a II given by Tits' formula does not contain the Dixon algebra T with a 3 representation of der(H) Theorem: The algebra a II given by Tits' formula does not contain the Dixon algebra T = C ⊗ H ⊗ O when H ⊂ T corresponds to the representations 3 ⊕ 1 with respect to der(H) = su 2 .Proof: By applying Tits' formula (with Barton-Sudbery's modification), one obtains 2 (C ⊗ H) = so 7 ⊕ so 6 ⊕ 3 • (7, 4) = g 2 ⊕ so 6 ⊕ 3 • (7, 4) ⊕ (7, 1) . ( We also recall that Let us now find all su 2 subalgebras of so 6 in (194): 1.

5
Projective lines over C ⊗ O via C ⊗ J 2 (O) 5.1 Minimal left ideals of Cl(6) via chain algebra C ⊗ ← − O
2 (C ⊗ H) are non-real, being of the form d = d 1 + Iid 2 + Ijd 3 + Ikd 4 , with d 1 , d 2 , d 3 , d 4 ∈ R, and 4 4.2 Generalized minimal left ideals of C ⊗ J 2 (H)To build up to projective lines of C ⊗ H ⊗ O, the physics of spinors for C ⊗ H are uplifted to C ⊗ J 2 (H).The C ⊗ H spinors are also embedded into C ⊗ J 2 (H) by placing ψ D in the upper-right component and adding by its quaternionic Hermitian conjugate to obtain an element of C ⊗ J 2 (H), (2,1,16) also contains degrees of freedom for right-chiral antiparticles with opposite charges via(1,2,16), which leads to a physicist's convention to ignore writing down the conjugate representation.Each of the 16 Weyl spinors is an element of C 2 .When working with C ⊗ H ⊗ ← − O , there are no two-component vectors, so it is necessary to find two copies of 16.When Furey explored Cl(10)from C⊗H⊗ ← − O , a 16 with its conjugate representation was found, instead of two 16's to give(2,1,16)for a single generation of Standard Model fermions.This led to the so-called fermion doubling problem.
(10)1)for the Standard Model.In this manner, C ⊗ H ⊗ O allows for Spin(10)to act from the left.While the full Cl(4) spacetime algebra cannot be found, the remaining right action remarkably picks out SL(2, C) as SU (2) C .A collection of left-chiral Weyl spinors in the (2, 1, 16) representation of SL(2, C)× 3 Uplift to O ⊗ J 2 (C ⊗ H) Next, we seek to obtain the physics of C ⊗ H ⊗ O by uplifting to O ⊗ J 2 (C ⊗ H).The Hermitian conjugate of O ⊗ J 2 (C ⊗ H) takes conjugation with respect to both C and H. Uplifting an element f ∈ e 61 − e 25 ), 26 + e 15 − e 37 ).