Three generations of colored fermions with S 3 family symmetry from Cayley–Dickson sedenions

An algebraic representation of three generations of fermions with SU ( 3 ) C color symmetry based on the Cayley–Dickson algebra of sedenions S is constructed. Recent constructions based on division algebras convinc-ingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difﬁcult to substantiate. We motivate S as a nat-ural algebraic candidate to describe three generations with SU ( 3 ) C gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of C (cid:2)( 6 ) , generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the S 3 automorphism of order three, which is an automorphism of S but not of O , to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.


Introduction
Despite its great practical success in colliders and other experiments, there are several unexplained features of the Standard Model of particle physics (SM) which lack a deeper theoretical motivation.These include, among others, a derivation of the SM gauge group from first principles, an explanation for why some representations of the SM gauge group correspond to particle multiplets whereas others do not, and an account for why fermions come in three generations.These theoretical shortcomings may be suggestive that the SM ultimately emerges from a more fundamental physical principle or mathematical structure.
In an attempt to establish the geometric and algebraic roots of the SM, several proposals have been put forth over the years which take as its essential mathematical ingredients (tensor products of) the only four normed division algebras over the reals: R, C, H, and O. Instead of unifying the internal symmetries into a single larger group, as is done in grand unified theories (GUTs) such as SU (5) and Spin (10), these division algebraic approaches attempt to unify the gauge groups together with the leptons and quarks that they act on into a single unified algebraic framework, in terms of an algebra acting on itself.
The octonions O, the largest of the division algebras, were first considered in the 70s for their intriguing efficacy in describing quark color symmetry [1].Dixon [2,3,4] considers the algebra R ⊗ C ⊗ H ⊗ O and its invariant subspaces in connection to the particles and charges of the SM.The algebra R ⊗ C ⊗ H ⊗ O has exactly the right dimensions (32 complex) to describe one generation of fermions.In a closely related approach, Furey studies the minimal ideals of the Clifford algebras Cℓ(4), and Cℓ (6), generated from C ⊗ H, and C⊗O respectively [5,6].In her approach, the leptons and quarks correspond to elements of these minimal ideals, and the gauge symmetries are those unitary symmetries that preserve the ideals.In particular, the part C ⊗ O part of Dixon's algebra can be associated to the color and electric charge internal degrees of freedom, with the color gauge group SU (3) corresponding to the maximal compact subgroup of the exceptional group G 2 of automorphisms of the O which fixes one of the octonion units.
Existing division algebraic models offer an elegant algebraic construction for the internal space of a single generation of leptons and quarks.Despite several attempts [4,33,34], a clear algebraic origin for the existence of three generation is yet to be found.The Pati-Salam model, as well as both the SU (5) and Spin (10) grand unified theories likewise correspond to single generation models, lacking any theoretical basis for three generations, which ultimately has to be imposed by hand.
Furey identifies three generations of color states directly from the algebra Cℓ(6) generated from the adjoint actions of C ⊗ O [33].The algebra Cℓ( 6) is 64 complex dimensional.Constructing two representations of the Lie algebra su(3) within this algebra, the remaining 48 degrees of freedom transform under the action of the SU (3) as three generations of leptons and quarks.The most obvious extension to include U (1) em via the number operator, which works in the context of a one-generation model, fails to assign the correct electric charges to states.A generalized action that leads to a generator that produces the correct electric charges for all states is introduced in [35].
Dixon on the other hand considers the algebra in order to represent three generations, with a single generation being described by T 2 , a complexified (hyper) spinor in 1+9D spacetime [3].However, the choice T 6 , as opposed to any other T 2n appears rather arbitrary, although can be motivated from the Leech lattice.
These division algebraic models share many similarities with those based on the exceptional Jordan algebra J 3 (O) consisting of three by three matrices over O, which has likewise been proposed to describe three generations [27,28,29,30,32,31].In these models, each of the three octonions in J 3 (O) is likewise associated with one generation via the three canonical J 2 (O) subalgebras of J 3 (O).
In [25] it is argued that R ⊗ C ⊗ H ⊗ O-valued gravity can naturally describe a grand unified field theory of Einstein's gravity with a Yang-Mills theory containing the SM, leading to a SU (4) 4 symmetry group that potentially extends the SM with an extra fourth family of fermions.The existence of a fourth generation of fermions lacks experimental support however.
In [36] it is shown how, by choosing a privileged C subalgebra of O, it is possible to reduce ten dimensional spacetime represented by SL(2, O) to four dimensional spacetime SL(2, C).This process of dimensional reduction naturally isolates three H subalgebras of O: those that contain the privileged C subalgebra.These three intersecting H subalgebras are subsequently interpreted as describing three generations of leptons.
Starting with R, each of the remaining three division algebras can be generated via what is called the Cayley-Dickson (CD) process.This process does not terminate with O however, but continues indefinitely to produce a series of 2 n -dimensional algebras.We therefore ask the question: Can we go beyond the division algebras, to the CD algebra of sedenions S, generated from O, in order to describe three generations?
The present paper advocates that the CD algebra of sedenions S constitutes a natural mathematical object which exhibits the algebraic structure necessary to describe the internal space of three generations.We restrict ourselves for the time being to considering only the SU (3) C color symmetry of leptons and quarks.
The algebra S was first proposed to play a role in describing three generations in [34].The key idea behind that proposal was to generalize the constructing of three generations of leptons in terms of three H subalgebras of O in [36] to three generations in terms of three O subalgebras of S, where each generation is associated with one copy of O.One finds, as in [36], that the resulting three generations are not linearly independent.It was suggested in [34] (and later in [14]) that this overlap could provide an algebraic basis for neutrino oscillations and quark mixing, although the viability of this idea remains to be investigated.
The model in [34] suffers from two significant drawbacks.Each generation comes with its own copy of SU (3) C thereby also requiring three generations of gluons, for which there is currently no experimental evidence.Additionally, Aut(S) = Aut(O) × S 3 , where Aut(O) = G 2 , and S 3 is the permutation group of three objects [37,38].The S 3 automorphisms of S were however not given any clear physical interpretation, in part because these automorphisms stabilize the octonion subalgebras in S.
The model we presented here builds on [34] and seeks to resolve the shortcoming just mentioned.Instead of associating each O subalgebra of S with one generation, we use all three O subalgebras to construct a single generation.This corresponds to a direct generalization of the construction in [5] where three H subalgebras of O are used to construct a single generation.Subsequently, we utilize sedenion S 3 automorphism of order three to generate the additional two generations.This construction provides a clear interpretation of the new S 3 automorphism.Furthermore, all three generations transform as required under a single copy of the gauge group SU (3) C , thereby avoiding introducing three generations of gluons.
In the next section we provide a brief overview of the normed division algebras, in particular the quaternions H and octonions O.In Section 3 we review the construction of one generation of fermions with unbroken SU (3) c × U (1) em gauge symmetry from C ⊗ O, following closely [5].The Cayley-Dickson construction and the algebra of sedenions are discussed in Section 4. Finally we present our three generation model based on the algebra of sedenions in Section 5. We conclude with an outlook of how to develop the model further, and a discussion.

Normed division algebras
A division algebra is an algebra over a field where division is always well-defined, except by zero.A normed division algebra has the additional property that it is also normed vector spaces, with the norm defined in terms of a conjugate.A well-known result by Hurwitz [39] is that there exist only four normed division algebras (over the field of real numbers): R, C, H, O, of dimensions one, two, four and eight respectively.In going to higher-dimensional algebras, successive algebraic properties are lost: R is self-conjugate, commutative and associative, C is commutative and associative (but no longer self-conjugate), H is associative but no longer commutative, and finally O is neither commutative nor associative (but alternative).
The quaternions H are a generalization of the complex numbers C with three mutually anticommuting imaginary units I, J, K, satisfying I 2 = J 2 = K 2 = IJK = −1, which implies IJ = K = −JI, JK = I = −KJ, and KI = J = −IK.A general quaternion q may then be written as With the quaternion conjugate q defined as q = q 0 − q 1 I − q 2 J − q 3 K.The norm of a quaternion |q| is subsequently defines by |q| 2 = qq = qq, and the inverse q −1 = q/|q| 2 .The automorphism group of H is SU (2).Indeed, there is an isomorphism between the quaternions H and the real Clifford algebra Cℓ(0, 2), while the complexified quaternions C ⊗ H (isomorphic to the Pauli algebra) are isomorphic to the complex Clifford algebra Cℓ(2).Note, however, that C ⊗ H is not a division algebra (but remains associative), and manifestly contains projectors, for example: The octonions O are the largest division algebra, of dimension eight.Its orthonormal basis comprises seven imaginary units: i 1 , ...i 7 , along with the unit 1 = i 0 .A general octonion x may then be written as with the octonion conjugate x defined as The norm of an octonion |x| is subsequently defines by |x| 2 = xx = xx, and the inverse The multiplication of octonions 1 is captured in terms of the Fano plane Fig. 1.Each projective line in the Fano plane corresponds (together with the identity i 0 ) to an H subalgebra; there are seven such subalgebras.Like with H, all the imaginary units anticommute under multiplication.Unlike with H, the multiplication of elements not belonging to the same H subalgebra is non-associative.For example i 4 (i 7 i 6 ) = −i 5 ̸ = i 5 = (i 4 i 7 )i 6 .Octonion multiplication however is alternative x(xy) = (xx)y and y(xx) = (yx)x, ∀x, y ∈ O.The complexified octonions C ⊗ O are again not a division algebra (but remains alternative).
As vector spaces O = C 4 .The splitting of O as C ⊕ C 3 relies on choosing a preferred octonion unit i a (and hence a preferred C subalgebra in O).For our purpose we choose i 4 .The map [36] 1 There are different multiplication rules for O used by different authors in the literature.Here we follow the multiplication table used in [40] Figure 1: The Fano plane, encoding the multiplicative structure of our octonions, where a ≡ i a , a = 1, ..., 7. Note that each line is cyclic, representing a quaternionic triple.
where ī4 indicates the octonion conjugation, then projects O down to this preferred C ⊂ O, and we can write the octonion x = x 0 i 0 + ... + x 7 i 7 as: Note that the product i 4 x ī4 is defined unambiguously since O is alternative.The automorphism group of O is the 14-dimensional exceptional Lie group G 2 .This exceptional group contains SU (3) as one of its maximal subgroups, corresponding to the stabilizer subgroup of one of the octonion imaginary units, or equivalently, the subgroup of Aut(O) that preserves the representation of O as the complex space C ⊕ C 3 .This splitting is associated with the quark-lepton symmetry [36].The space of internal states of a quark is then the three complex dimensional space C 3 whereas the internal space of a lepton is C.
Since O and C ⊗ O are nonassociative, they are not representable as matrix algebras (with the standard matrix product).The algebra generated from the composition of left and right actions of O (and C ⊗ O) however is associative, since each such left (right) action corresponds to a linear operator (endomorphism).
Let L a (R a ) denote the linear operator of left (right) multiplication by a ∈ C ⊗ O: Then The mappings a → L a and a → R a do not correspond to algebra homomorphisms as they each generate an associative algebra called the associative multiplication algebra2 .They do however preserve the quadratic relations ⟨x, y⟩ Since L a (R a ) correspond to linear operators, they can be represented as 8 × 8 complex matrices (acting on the vector space C ⊗ O written as a column vector).
Due to the nonassociativity of O, the left (right) associative multiplication algebra of C ⊗ O contains genuinely new maps which are not captured by C ⊗ O.For example, i 3 (i 4 (i 6 + i 2 )) ̸ = y(i 6 + i 2 ) for any y ∈ C ⊗ O.There are a total of 64 distinct left-acting complex-linear maps from C ⊗ O to itself, and these (due to the given identities above) provide a faithful representation of Cℓ (6).
Denoting the 64-dimensional left (right) associative multiplication algebra generated from left (right) actions of C ⊗ O on itself by (C ⊗ O) L ((C ⊗ O) R ), one finds that any left (right) action can always be rewritten as a right (left) action [4].That is: This is in contrast to a similar construction for C ⊗ H, where one finds that the left and right actions are genuinely distinct, each generating a copy of Cℓ(2).The left and right adjoint actions in this case commute, and only by considering both does one obtain a basis for M at(4, C) ∼ = Cℓ(4).
3 One generation of electrocolor states from C ⊗ O Let e 1 := L i1 , ..., e 6 := L i6 be a generating basis (over C 6 ) for Cℓ(6) associated with the left multiplication algebra of the complex octonions, satisfying e 2 i = −1, e i e j = −e j e i .Define the Witt basis Here † corresponds to the composition of complex and octonion conjugation.This new basis satisfies the anticommutation relations Each pair of ladder operators in isolation generates Cℓ(2), and is associated with one of the three H subalgebras of O that contain the privileged complex subalgebra generated by i 4 .Subsequently, we obtain three anticommuting copies of Cℓ(2), which when considered together generate the full Cℓ(6) left multiplication algebra of C ⊗ O.
The unitary symmetries that preserve the Witt basis, and hence the minimal left ideals is U (3) = SU (3) × U (1).The generators of this symmetry, written in terms of the Witt basis, are: The basis states of minimal ideals transform as 1 ⊕ 3 ⊕ 3 ⊕ 1 under SU (3), and this symmetry can therefore be associated with the color symmetry SU (3) C , justifying the choice of coefficients.The U (1) generator Q, related to the number operator Q = N/3, on the other hand gives correct electric charge for each state.The ideal S u contains the isospin up states, whereas the S d contains the isospin down states.One generation of leptons and quarks with correct unbroken SU (3) C × U (1) em symmetry can therefore be elegantly represented in terms of two minimal left ideals of Cℓ(6) generated from C ⊗ O.The dimension of the minimal ideals dictates the number of distinct physical states, whereas the gauge symmetries are those unitary symmetries that preserve the ideals (or equivalently, the Witt basis).

The Cayley-Dickson construction and the algebra of sedenions
The CD process is an iterative construction that generates at each stage an algebra (with involution) of dimension twice that of the previous.Each algebra is constructed as a direct sum of the previous algebra, so that C = R ⊕ Ri where i is the complex structure introduced in the process.Similarly, H = C ⊕ CJ, where i, J and iJ are identified with the quaternion imaginary bases I, J, K, and similarly O = H ⊕ Hi 4 .
This process does not terminate with O but continues, generating a series of 2 n -dimensional (non division) algebras.A generic element of the CD algebra A n , can then be written as a + bu, where a, b ∈ A n−1 , and u is the new imaginary unit introduced by the CD process applied to A n−1 .The fifth CD algebra A 4 (A 0 = R), generated from O, is the 16-dimensional algebra of sedenions S.This algebra is non-commutative, nonassociative, and not even alternative (x(xy) ̸ = (xx)y and y(xx) ̸ = (yx)x in general).Other properties, like flexibility ((xy)x = x(yx)) and power-associativity (x n associative), still hold (and hold for all CD algebras).
Because S is not a division algebra, it contains zero divisors.These are elements of the form There are 84 such zero divisors, and the subspace of zero divisors of unit norm is homeomorphic to G 2 [43].

Octonion subalgebras inside the sedenions
Let us now use {e 0 , e 1 , ..., e n 2 −1 } to denote an orthonormal basis for A n , so that Consider A 2 = H with basis {e 0 , e 1 , e 2 , e 3 }.There are three subalgebras isomorphic to C within H, each containing the identity and one of the imaginary units of H.These subalgebras correspond to three different complex structures in H, and the common intersection of these three C subalgebras is isomorphic to R. The automorphism group of H is Aut(H) = SU (2).The subset of automorphisms of H that preserve a given complex structure is U (1), corresponding to the element wise stabilizer subgroup of SU (2).
Applying the CD process to H generates O with basis {e 0 , e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , e 7 } where e 4 is the newly introduced anticommuting imaginary unit and e i e 4 = e i+4 .Via this same construction, each of the three C ⊂ H subalgebras generate a quaternion: The common intersection of these three where this H is generated by {e 0 , e 4 , e 8 , e 12 }.O 1 , O 2 , O 3 are the only (octonion) subalgebras of S that contain e 4 , e 8 , and e 12 .Together with the identity, this corresponds to a quaternionic structure.
In addition to the eight octonion subalgebras of S, there are also a further seven quasi-octonion subloops Õ, satisfying all the same properties of the octonion subalgebras, except for the Moufang identities 5 .As such, they are not isomorphic to the octonion subalgebras.None of the Õ contain the element e 8 .

The left multiplication algebra of C ⊗ S
Despite the algebra S being non-associative and non-alternative, just as for O, we can consider the left actions of S on itself as linear operators generating an associative algebra.
The generalisation from (C ⊗ O) L to (C ⊗ S) L is not immediately obvious because the identities which held for (C ⊗ O) L , namely ...
However, since the linear operators corresponding to each complex sedenion left multiplication can be written as a 16 × 16 matrix with complex entries acting on C ⊗ S written as a column vector, one would expect to be able to generate M at(16, C) ∼ = Cℓ (8).It then remains to find a suitable set of sedenion elements that generate Cℓ(8) via their left multiplication.Closer inspection reveals that all the left multiplications of the original octonion elements e 0 = i 0 = s 0 , ..., e 7 = i 7 = s 7 do satisfy the identities ( 29) and ( 30 The left action of e 8 however anti-commutes with the left action of every other basis element.One possible generating basis for Cℓ( 8) is therefore given by the left multiplications of {e 1 , ..., e 8 }.Another possible generating basis is {e 8 , ..., e 15 }.Accepting some abuse of notation, we now simple write L ei = e i and take e 0 , e 1 , ..., e 8 as our generating basis for Cℓ (8).The left action of the remaining sedenion basis elements e 9 , ..., e 15 then need to be expressed as the left action of some element of Cℓ (8).After some trial and error, one finds that where e 123458 = e 1 e 2 e 3 e 4 e 5 e 8 = L e1 L e2 L e3 L e4 L e5 L e8 , e 18 = e 1 e 8 = L e1 L e8 etc are elements of Cℓ(8).

Automorphisms of sedenions
Schafer [37] showed that for CD algebras A n with n ≥ 4 (A 0 = R), the derivation algebra der(A n ) consists of derivations of the form a + bu → aD + (bD)u, where a, b ∈ A n−1 , u is the new anticommuting imaginary unit u introduces in the CD construction of A n from A n−1 , and D is a derivation of A n−1 .Brown [38] demonstrated that if θ ∈ Aut(A n−1 ), then are automorphisms of A n .Here * denotes conjugation in A n−1 , and ϵ and ψ, satisfying It follows that SU (2) × S 3 are automorphisms of O, but crucially, these are not all of the octonion automorphisms: SU (2) × S 3 ⊂ G 2 .However, for the cases where n = 4, 5, 6, the equality holds [38] In particular, this means that: Explicitly, the automorphisms of S are therefore give by where a, b ∈ O.The explicit action of ψ on the sedenion basis elements can be written as: where i = 1, ..., 7. The automorphism ψ corresponds to a simultaneous rotations in the seven e i − e i+8 planes by 2π/3, and therefore does not correspond to an automorphism of C ⊗ O.It is also possible to write the S 3 automorphisms in matrix form: ϵ : The fundamental symmetries of S are the same as those of O, although one find an additional S 3 symmetry, suggesting a threefold multiplicity of the automorphisms of O.
From the action of θ ′ on the sedenion units above it is immediately clear that e 8 is stabilized by the G 2 automorphisms.Furthermore, these G 2 automorphisms map e i , i < 8 to e j , j < 8, and e i+8 to e j+8 .They therefore do not mix the new sedenion elements e i+8 with the original octonion elements e i , i = 1, ..., 7.Only the S 3 automorphism psi of order three mixes the original octonion units with the new sedenion units.
Given that the stabilizer of e 4 in G 2 is SU (3), and e 8 is likewise an SU (3) singlet (as it is fixed by G 2 ), it follows that the SU (3) subgroup of G 2 fixes the entire quaternion generated by {e 0 , e 4 , e 8 , e 4 e 8 = e 12 }.The S 3 automorphisms on the other hand do not fix this quaternion, although they do stabilize it.Note that this quaternion corresponds precisely to the common intersection of our previously isolated octonion subalgebras O i , see eqn.(27).
5 Three generations of color states from C ⊗ S

Why not sedenions?
Since one generation of electrocolor states are efficiently represented starting from C ⊗ O, one might ask if S (or rather C ⊗ S) is an appropriate larger algebraic structure capable of describing three generations.Not being a division algebra is not grounds to disqualify S, for we point out that neither C⊗O nor Cℓ(6) generated as the left multiplication algebra are themselves division algebras.Furthermore, the algebra R⊗C⊗H⊗O is, like S, not even alternative.in fact, the construction of invariant subspaces (minimal ideals) relies explicitly on the use of projectors (and nilpotents), which altogether do not exist in division algebras.
There are several natural reasons to suspect that S exhibits the algebraic structure necessary to describe three full generations: 1. Aut(S) = Aut(O) × S 3 , and one finds a threefold multiplicity of the symmetries associated with O, 2. The process in Section 4 shows how to naturally isolate thee O subalgebras within S, which could perhaps be used to construct three generations, 3. The group Spin(8) generated from Cℓ(8) admits a triality, which has on occasion been suggested as a potential source of three generations [32,44].The group of outer-automorphisms of Spin( 8) is precisely S 3 .
One approach to construct three generations with SU (3) C symmetry is to use each of the three C⊗O i ⊂ S to generate (via its left action on itself, but not as a left action of C ⊗ S!) a Cℓ(6) algebra, and subsequently representing three generations in terms of the minimal left ideals of these three Cℓ (6).This approach was considered in [34], as a generalization of the construction of three generations of leptons from three H ⊂ O developed in [36].
There are several drawbacks to this approach however.Each generation requires its own copy of SU (3) resulting in three generations of gluons.Finally, the physical interpretation of the S 3 automorphisms remains obscured, because these automorphisms stabilize the (non-principle) octonion subalgebras in S.
The approach pursued here is different and seeks to resolve these shortcomings.We instead use each C ⊗ O i ⊂ S to construct a pair or fermionic ladder operators that each generate Cℓ(2), via their left multiplication action on all of C ⊗ S (instead of just C ⊗ O i ).The three pairs of ladder operators are independent of one another and hence we identify a single copy of Cℓ(6) ∼ = Cℓ(2) ⊗Cℓ( 2) ⊗Cℓ( 2), corresponding to a subalgebra of Cℓ (8).Thus, we will employ all three O i ⊂ S in order to construct a single generation of states.This corresponds to a direct generalization of the construction of reviewed in Section 3 where three H subalgebras of O are used to construct a single generation of color states.Subsequently, the order three S 3 automorphism of S will be used to generate two additional generations.This construction will therefore provide a clear interpretation of the new S 3 automorphism ψ.All three generations constructed in this manner transform as required under a single copy of the gauge group SU (3) C , thereby avoiding introducing three generations of gluons.
For each O i subalgebra, we define a single pair of raising and lowering operators as follows: It is readily checked that A i (A i w) = A † i (A † i w) = 0 and A i (A j w) = −A j (A i w), ∀w ∈ C ⊗ S, and therefore these ladder operators satisfy (as left actions on a general w ∈ S) the usual anticommutation relations: Each A i ∈ O i ⊂ S in a generalization of α i ∈ H i ⊂ O where now each ladder operators consists of four terms instead of just two.Subsequently we can proceed to construct two minimal left ideals, in a manner identical as in Section 3.These ideals are identical to S u and S d above, and (as will be demonstrated shortly) preserved by the same unitary symmetries, but with both the states and symmetry generators written in terms of the generalised ladder operators A i and A † i .
to represent chiral SU (2) L states in terms of two four-dimensional minimal ideals.The resulting model is then based on Cℓ(10) [6,47,48].In our present construction, we have restricted ourselves to a Cℓ(6) subalgebra of the full Cℓ(8) left multiplication algebra.The thus far unused Cℓ(2) could be combined with the right actions of the nilpotents ω i and ω † i to generate Cℓ(4) without the need to invoke H.This approach is currently being investigated by the authors.
It is well known that C ⊗ H ∼ = Cℓ(2) ∼ = SL(2, C).By complementing the present model sedenion model with a factor of H, it should be possible to include the Lorentz (spacetime) symmetries into the model.Specifically, it was shown in [5] that Weyl-, Dirac-and Majorana-spinors can all be represented in terms of the ideals of C ⊗ H ∼ = Cℓ(2).The spacetime symmetries would then be identical for all three generations, a desirable feature.
By choosing a complex structure within O, the ten dimensional spacetime described by SL(2, O) is reduced to four dimensional spacetime SL(2, C) [36,49].It would be interesting to see whether SL(2, O) spacetime can be extended to an 18 dimensional spacetime SL(2, S) which is then broken to SL(2, H) via our quaternionic structure inside S, and subsequently to SL(2, C) by choosing a complex structure inside H.
The construction of three generations presented here bears some intriguing resemblance to the three generation construction in [33,35].In our case, all three generations reside in a Cℓ(6) subalgebra of Cℓ (6).
Likewise in [33], the 48 degrees of freedom that remain in Cℓ( 6) once two representations of the Lie algebra su(3) have been accounted for are shown under the action of SU (3) to transform as three generations of leptons and quarks.The number operator in this construction no longer assigns the correct electric charges to the states however, an issue that was overcome in a later paper [35].How to include U (1) em into the sedenion model, as well as a detailed understanding of how these two complementary constructions are related remains to be worked out.
Likewise, it remains to be investigated in detail how the sedenion model proposed here relates to the constructions of three generations based on the exceptional Jordan J 3 (O), in which each of the three octonions in J 3 (O) is associated with one generation via the three canonical J 2 (O) subalgebras of J 3 (O) [27,28,29,30,31,32].
Given the richer algebraic structure provided by S, it may be possible to describe additional features of the SM.The S 3 automorphism of order three not only generates two additional generations, but also facilitate transformations between physical states of different generations.This suggests that S 3 could perhaps be used as a basis for including quark mixing and neutrino oscillations.This is something that has not yet been considered within the context of division algebras.It is interesting that several authors have proposed S 3 extensions of the SM to explain the hierarchy of quark masses and mixing in the SM [50,51,52,53,54].

Discussion
We have argued that the CD algebra S provides a suitable algebraic structure to describe three generations.Our focus has been restricted to the SU (3) C color gauge symmetry.Three intersecting O subalgebras of S are used to construct a Witt basis for Cℓ (6).Two minimal left ideals are then use to represent one generation of electrocolor states.Subsequently, the S 3 automorphism of order three of S is then applied to the Cℓ(6) Witt basis in order to obtain exactly two additional generations of color states, but not electrocolor states.
In [4,5], Cℓ(6) arises as the left multiplication algebra of C ⊗ O. Instead, in our approach Cℓ(6) does not arise as the multiplication algebra of a single octonion algebras, but rather from three intersection octonion subalgebras of the sedenions, with each octonion subalgebra contributing a Cℓ(2) factor.The model presented here overcomes two limitations of a previous three generation model based on sedenions.First, only a single copy of SU (3) C correctly transforms the states of all three generations.We thereby avoid introducing three generations of gauge bosons.Additionally in the present model, the order three S 3 automorphisms of S if given a clear physical interpretation as a family symmetry, responsible for generating two additional generations from the first.
One compelling reason to consider division algebras as the foundational mathematical input from which to generate SM particle multiplets and gauge symmetries, is that there are only four of them.Since the CD process generates an infinite series of algebras, one might question whether going beyond the division algebras and including S is a wise idea, or if it opens the door to considering ever larger algebras.The derivation algebra for all CD algebras A n , n ≥ 3 is equal to g 2 however [37].Furthermore, at least for the cases n = 4, 5, 6, the automorphism group of each successive CD algebra only picks up additional factors of S 3 [38].It therefore seems unlikely that CD algebras beyond A 4 = S will provide additional physical insight.As an interesting aside, the sphere S 15 , associated with the imaginary (pure) sedenions, is the largest sphere to appear in any of the four Hopf fibrations.13

Table 1 :
Sedenion multiplication table, generated from e 1 , e 2 , e 4 and e 8 corresponding to the new C, H, O and S imaginary units respectively.