Left-Right Symmetric Fermions from Complex Split Biquaternions and Bioctonions

Introduction

The Standard Model treats the two chiralities of the fermions unequally: the left-handed components sit in $\mathrm{SU}(2)_L$ doublets and the right-handed components in singlets, and no right-handed neutrino is required at all. The left-right symmetric models of Pati and Salam and of Mohapatra and Pati restore the symmetry by enlarging the electroweak factor to $\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$, so that every left-handed doublet has a right-handed partner and the right-handed neutrino becomes a member of an $\mathrm{SU}(2)_R$ doublet — a gauge singlet of the Standard Model, and therefore sterile, but a state of the larger theory. A construction of Vaibhav and Singh builds exactly this pairing out of a pair of division-algebra Clifford algebras: the complex split biquaternion algebra $\mathrm{Cl}(3)$ for the leptons and the complex split bioctonion algebra $\mathrm{Cl}(7)$ for the quarks, each splitting into two copies of a smaller algebra — $\mathrm{Cl}(2)$ and $\mathrm{Cl}(6)$ respectively — which carry the two chiralities and are interchanged by a parity automorphism. This article states the algebraic core of that construction and sorts its steps into what is established, what is read, and what is left open.

The algebraic core is proved in the two companion mathematics articles. The complex split biquaternions and the doubling $$ \mathrm{Cl}(3) \cong \mathrm{Cl}(2)\oplus\mathrm{Cl}(2) \cong \mathbb{B}\oplus\mathbb{B} $$ are Complex Split Biquaternions and the Clifford Algebra Cl(3); the complex split bioctonions and the doubling $$ \mathrm{Cl}(7) \cong \mathrm{Cl}(6)\oplus\mathrm{Cl}(6) $$ are Split Bioctonions and the Clifford Algebra Cl(7); and the octonionic chain algebra $\mathrm{Cl}(6)\cong M_8(\mathbb{C})$, with its ladder operators, its intrinsic $\mathrm{su}(3)\oplus\mathrm{u}(1)$, its primitive idempotent and its charge spectrum $0,\tfrac13,\tfrac13,\tfrac13,\tfrac23,\tfrac23,\tfrac23,1$, is Complex Octonions and the Clifford Algebra Cl(6), whose physics reading is The Particle Roster of the Complex Octonionic Chain Algebra. The neutrino content against which the construction is compared is The Neutrino and Majorana Fermions in Biquaternionic Form and The Seesaw Mechanism and Majorana Mass in Biquaternionic Form; the anomaly conditions that decide whether a sterile state is allowed are Anomalies and Anomaly Cancellation in Biquaternionic Form; the framework's own ledger on the electroweak factor, on the generation count and on the ceiling it cannot pass is The Standard Model under the Biquaternion Framework — A Research Agenda, The Number of Generations and the Biquaternion Algebra and Grand Unification and the Biquaternion Algebra Ceiling; and the exceptional structures of the source's further programme are Octonions and the Exceptional Lie Groups.

As in The Particle Roster of the Complex Octonionic Chain Algebra, the construction is outside the biquaternion algebra: $\mathbb{B}\cong M_2(\mathbb{C})$ is eight-dimensional and real, the chain algebras here are $64$- and $128$-dimensional and complex, and no operation of $\mathbb{B}$ reaches them. The article is in the physics category as a record of the literature's most economical answer to the chirality and sterile-neutrino questions, not as a derivation inside the framework, and no result of it feeds back into the framework's ledger. Besides the doubling and its reading, the record covers the further pieces of the source that later passes added: the explicit construction by which each of the two weak factors is made to act on one chirality only, with the part of it recomputed here marked; the reading of the volume element as the Higgs; the source's proposal that the two chiralities existed in a combined leptoquark state before symmetry breaking; the source's later programme, in which the right-handed weak factor becomes a chiral formulation of gravity in a ten-dimensional octonionic pre-spacetime and the second summand's charge becomes a square root of mass; and the recomputation, in the companion mathematics article, of the $\mathrm{Cl}(5)$ statement that a second pass had left as the source's.

The Two Algebraic Facts

Two structural facts, both proved in the mathematics articles, are all that the construction's treatment of chirality uses.

The doubling. Each of the two algebras has a central volume element which is an involution. For $\mathrm{Cl}(3)$ the volume element is $\omega=E_1E_2E_3$ with $\omega^2=1$, and for $\mathrm{Cl}(7)$ it is $J=e_1\cdots e_7$ with $J^2=1$; in both cases the idempotents $\Pi_\pm=\tfrac12(1\pm\omega)$ and $\Pi_\pm=\tfrac12(1\pm J)$ are a complete family of orthogonal central idempotents, and the algebra is the direct sum of two copies of the smaller algebra, $$ \mathrm{Cl}(3) \cong \mathrm{Cl}(2)\oplus\mathrm{Cl}(2), \qquad \mathrm{Cl}(7) \cong \mathrm{Cl}(6)\oplus\mathrm{Cl}(6). $$ The parity automorphism $\sigma:e_i\mapsto-e_i$, fixing the even elements and negating the odd ones, sends $\omega\mapsto-\omega$ and $J\mapsto-J$ and thereby interchanges the two copies. The copies are the two pinor representations, inequivalent, distinguished by the eigenvalue $\pm1$ of the central volume element. That $\omega$ is a genuine exchange operator was checked here in the faithful representation $\mathrm{Cl}(0,3)\cong\mathbb{H}\oplus\mathbb{H}$: with $e_1\mapsto(i,i)$, $e_2\mapsto(j,j)$ and $e_3\mapsto(k,-k)$, the left set $(1,e_1,e_2,e_1e_2)$ is the copy whose two $\mathbb{H}$ components are equal, the right set $\omega(1,-e_1,-e_2,-e_1e_2)$ is the copy whose components are opposite, and multiplication by $\omega=e_1e_2e_3$ maps each copy onto the other, with $\omega^2=1$, $\omega^{*}=\omega$ and $\omega$ central (residuals $0$).

The two $\mathrm{SU}(2)$ factors from the left and the right. On the biquaternion algebra $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ there are two commuting actions of the quaternion units, the left multiplications $L_{x}:y\mapsto xy$ and the right multiplications $R_{x}:y\mapsto yx$. The right multiplications compose the other way round, $R_xR_y=R_{yx}$, so they generate the opposite algebra; both the left and the right action of $\mathbb{H}$ generate a copy of $\mathbb{H}$, whose imaginary part is a Lie algebra isomorphic to $\mathrm{su}(2)$, and the two commute. Together the left and the right actions generate the full endomorphism algebra, $$ \operatorname{span}_{\mathbb{C}}\{\,L_xR_y\,\} = \operatorname{End}_{\mathbb{C}}(\mathbb{B}) \cong M_4(\mathbb{C}) \cong \mathbb{C}\mathrm{l}_4, $$ whose orthogonal part is $\mathrm{so}(4)\cong\mathrm{su}(2)\oplus\mathrm{su}(2)$. These statements were checked on the four-dimensional model of $\mathbb{B}$: the left and right actions commute identically, each spans a four-dimensional algebra, the left action reproduces the quaternion product $L_{e_1}L_{e_2}=L_{e_3}$ while the right action reverses it, and the span of the products $L_xR_y$ is sixteen-dimensional, i.e. all of $M_4(\mathbb{C})$.

The Left-Right Symmetric Reading

The two copies as the two chiralities. The doubling is the construction's answer to the chirality problem. Where the single chain algebra $\mathrm{Cl}(6)$ of The Particle Roster of the Complex Octonionic Chain Algebra carries one generation in one chirality and needs a second ideal plus an enlarged supercharge sector to become chiral, the doubled algebra carries the two chiralities as its two simple summands. The reading is that one summand is the left-handed sector and the other the right-handed sector, that the parity automorphism $\sigma$, which interchanges them, is the discrete left-right symmetry of the left-right symmetric model, and that the two summands carry the same intrinsic $\mathrm{su}(3)\oplus\mathrm{u}(1)$ and the same charge spectrum — the fact, proved in the mathematics article, that the ladder operators, the number operator and the $\mathrm{su}(3)$ generators all commute with the volume element $J$ — so that the two sectors differ in chirality and in nothing else. The states of one summand are

$$ 1(0),\quad 3(\tfrac13),\quad 3(\tfrac23),\quad 1(1), $$

the charge multiset of half a generation, and the summand's conjugate ideal supplies the antiparticles, exactly as in the undoubled construction of the particle-roster article. One trap has to be kept in view here. The chirality is carried by which set a state comes from and not by its expression: the source's own equations give the same element $\tfrac12(1+ie_1e_2)$ for the left-handed neutrino of the first set and the right-handed neutrino of the second, and the source says so explicitly. A reader who looks for two different algebraic expressions for the two chiralities will not find them, and the exchange operator that does distinguish the sets is the volume element $\omega$, treated below.

The weak factors from the two actions. The weak $\mathrm{SU}(2)$ factors come from a different place, and the two facts of the previous section have to be kept apart. The doubling supplies the two chiralities, and the two commuting quaternion actions of the biquaternion algebra supply two commuting $\mathrm{su}(2)$ algebras. The reading is that the left action is $\mathrm{SU}(2)_L$ and the right action is $\mathrm{SU}(2)_R$: the left multiplications act on the internal biquaternion index of the states, the right multiplications act on a separate one, and the two commute by the computation above, so that the states of the doubled construction can be organised into doublets of the left action and doublets of the right action without the two actions interfering. The isospin structure that the undoubled construction had to leave to a reading — the second non-abelian factor whose absence makes the minimal version non-chiral — is here supplied by the right action, and the electroweak factor of the enlarged theory is the $\mathrm{su}(2)\oplus\mathrm{su}(2)=\mathrm{so}(4)$ of the previous section. The source's mechanism is more specific: the right action of the quaternion units on the ideal is stated to annihilate the right-handed component while interchanging the two isospin states of the left-handed component, so that the two factors act on opposite chiralities, which is the defining statement of the left-right symmetric model. The corpus records the finite algebraic facts — the commutation of the two actions, the generation of $M_4(\mathbb{C})$, and the diagonalisation of the right action on the ideal — and marks the identification of the two actions with the two weak factors of the model as a reading, because the precise isospin alignment depends on the supercharge sector chosen and is not fixed by the algebra alone. The mechanism itself — which state the generators annihilate and which pair they interchange — is worked out below in Why Each Factor Acts on One Chirality Only, where its algebraic skeleton is recomputed and the part that remains a reading is named.

The right-handed neutrino. With the right-handed sector present, the state of the second summand of charge $0$ is available: a neutral, colourless, $\mathrm{SU}(2)_L$-singlet fermion, which is the right-handed neutrino $\nu_R$. In the Standard Model this state is not required — the fifteen states of a generation are the left-handed doublet, the right-handed singlets and their conjugates, and the right-handed neutrino can be dropped because it is a complete gauge singlet under $\mathrm{SU}(3)\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$. It is exactly because it is sterile that the construction's presence of it is conspicuous: a sterile fermion is unconstrained by the anomaly conditions, which act only on the gauge-charged content, so the doubling costs nothing at the level of gauge consistency and adds a state that the Standard Model's own ledger must otherwise introduce by hand for neutrino mass. This is the construction's sharpest contact with the framework's ledger: the framework's own count of the electroweak factor is $\mathrm{U}(2)$ and never $\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$, and its $\nu_R$ status is exactly the question this construction answers by construction rather than by addition.

The Pati–Salam reading. The full gauge group of the left-right symmetric unification is the Pati–Salam group $$ \mathrm{SU}(4)_c\times\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R , $$ in which the colour $\mathrm{SU}(3)_c$ and the baryon-minus-lepton number $\mathrm{U}(1)_{B-L}$ sit inside $\mathrm{SU}(4)_c$, and the two $\mathrm{SU}(2)$'s act on the two chiralities. The construction supplies the two $\mathrm{SU}(2)$'s from the two quaternion actions, and the $\mathrm{SU}(3)$ from the maximal totally isotropic subspace of the octonionic chain algebra in each summand; the $\mathrm{U}(1)_{B-L}$ is the abelian direction identified with the number-operator charge $Q=\tfrac13N$ modulo the colour structure. The assignment of the states to the Pati–Salam multiplets, in the source's reading, places one generation of quarks and leptons, with both chiralities, in the doubled algebra; the corpus does not recompute the multiplet assignment, and records it as a reading, in the same sense as the particle-roster article's assignment of its sixteen states.

The weak bosons as ladder operators. The source identifies the composite ladder operator $\Omega_L=\alpha_1\alpha_2\alpha_3$ of each summand with the charged weak boson: $\Omega_L$ has charge $-1$ and $\Omega_L^{\dagger}$ charge $+1$ under the number-operator charge, right multiplication by $\Omega_L$ changes a particle's isospin from up to down and right multiplication by $\Omega_L^{\dagger}$ reverses that on the antiparticles, so that $\Omega_L,\Omega_L^{\dagger}$ "mimic" $W^-$ and $W^+$. In the doubled algebra the same statement is made for the second summand's $\Omega_R,\Omega_R^{\dagger}$ as the right-handed analogues. This is a reading of the ladder operators that the chain-algebra article already uses — the corpus records it and notes that a ladder operator has the charge and the isospin action of a $W$, but that the construction contains no gauge field, no kinetic term and no coupling, so "mimics the $W$" is a correspondence of quantum numbers, not a derivation of the weak interaction. The source later abandons this assignment for the right sector in favour of using $\mathrm{SU}(2)_R$ for gravity, as recorded under the further programme below; the corpus keeps both statements with that shift marked.

Why Each Factor Acts on One Chirality Only

The reading above identifies the left action with $\mathrm{SU}(2)_L$ and the right action with $\mathrm{SU}(2)_R$ and leaves open the mechanism that makes each act on one chirality only. The source supplies it, and its algebraic skeleton can be recomputed.

The construction. With the octonionic ladder operators $\alpha_1,\alpha_2,\alpha_3$ and the idempotent generator $\Omega_L=\alpha_1\alpha_2\alpha_3$ of Complex Octonions and the Clifford Algebra Cl(6), the source takes the right action of $\mathbb{C}\otimes\mathbb{H}$ on $\mathbb{C}\otimes\mathbb{H}$ — the tensor product $(1,e_1,e_2,e_1e_2)\otimes_{\mathbb{C}}(1,e_1,e_2,e_1e_2)$ of the two quaternion bases, said to span $\mathbb{C}\mathrm{l}(4)$ — and builds four states from the totally isotropic elements $$ \beta_1=\tfrac12(-e_1+ie_2\tau_3),\qquad \beta_2=\Omega_L^{\dagger}ie_2 , $$ namely the idempotent $V_R=\beta_1^{\dagger}\beta_2^{\dagger}\beta_2\beta_1$ and its excitations $$ V_L=V_R\beta_1^{\dagger},\qquad e^-_R=V_R\beta_2^{\dagger},\qquad e^-_L=V_R\beta_1^{\dagger}\beta_2^{\dagger}, $$ with the isospin generators $$ T_a=\tau_a\,\frac{1+ie_1e_2}{2},\qquad \tau_1=\Omega_L+\Omega_L^{\dagger},\quad \tau_2=i\Omega_L-i\Omega_L^{\dagger},\quad \tau_3=\Omega_L\Omega_L^{\dagger}-\Omega_L^{\dagger}\Omega_L . $$ The source's statement is that the $T_a$ annihilate $V_R$ and $e^-_R$ while interchanging $e^-_L$ and $V_L$ as the two isospin states, so that $\mathrm{SU}(2)_L$ acts on the left-handed particles and their right-handed antiparticles; and that the mirror construction — the right action on $\omega\,\mathbb{C}\otimes\mathbb{H}$ rather than on $\mathbb{C}\otimes\mathbb{H}$ — reproduces the same picture with the roles exchanged, so that $\mathrm{SU}(2)_R$ acts on the right-handed particles and their left-handed antiparticles.

What was checked, and what stays a reading. Three ingredients were recomputed. First, $\Omega_L^2=0$ with $\{\Omega_L,\Omega_L^{\dagger}\}=1$ gives $\tau_3=1-2N$ with $N=\Omega_L^{\dagger}\Omega_L$, so on the two-dimensional module $\{V,\Omega_L^{\dagger}V\}$ the triple $\tau_1,\tau_2,\tau_3$ closes on $\mathrm{su}(2)$, $[\tau_a,\tau_b]=-2i\varepsilon_{abc}\tau_c$ — the standard relations with the orientation reversed, which is what the source's factor of $i$ in $\tau_2$ produces — and $\tau_1$ interchanges $V$ and $\Omega_L^{\dagger}V$. Second, $P=\tfrac12(1+ie_1e_2)$ is self-adjoint and idempotent, with $P(1-P)=0$ and $\operatorname{rank}P=2$. Third, $T_a=\tau_a\otimes P$ is a tensor product of two commuting factors, so the $T_a$ satisfy the same $\mathrm{su}(2)$ relations and satisfy $T_a(1-P)=0$ identically: the generators annihilate one component of the algebra and act only on the component $P$ selects. That identity is the algebraic content of "the generators annihilate $V_R$ and $e^-_R$ while interchanging $e^-_L$ and $V_L$" — which of the four named states lie in the selected component is the source's identification of states, but the selectivity itself is a property of the idempotent, not a reading. All residuals were $0$. The article therefore records the mechanism as established in its algebraic skeleton and read in its particle assignment: the commutation, the closure on $\mathrm{su}(2)$ and the annihilation of the complementary component are exact; the naming of $V_L,e^-_L$ as the isospin doublet and of $V_R,e^-_R$ as the annihilated states depends on the supercharge sector, which the algebra does not fix.

A parity operator from the right action. The source notes that the right action of $\mathbb{C}\otimes\mathbb{H}$ on $\mathrm{Cl}(3)$ is said to give $\mathrm{Cl}(5)$, whose Coxeter element is $\gamma_5=i\gamma_0\gamma_1\gamma_2\gamma_3$, so that for $\mathrm{Cl}(3)$ a parity reversal operator does not have to be introduced by hand; and that $\mathrm{Cl}(5)\cong\mathrm{Cl}(6)^+$ links this to Stoica's reading of the standard model from a pair of $\mathrm{Cl}(6)$'s, one carrying $\mathrm{SO}(4)$ and the other $\mathrm{SU}(3)\times\mathrm{U}(1)$. The corpus has verified the first half of the tensor statement — $\operatorname{span}_{\mathbb{C}}\{L_xR_y\}=\operatorname{End}_{\mathbb{C}}(\mathbb{B})\cong M_4(\mathbb{C})\cong\mathbb{C}\mathrm{l}_4$, which is the source's "the right action on $\mathbb{C}\otimes\mathbb{H}$ gives $\mathrm{Cl}(4)$" — and the $\mathrm{Cl}(5)$ statement has since been recomputed in the companion mathematics article Complex Split Biquaternions and the Clifford Algebra Cl(3), in its section The Right Action and the Enveloping Algebra: the algebra generated by the left action of $\mathrm{Cl}(3)$ and the right action of $\mathbb{C}\otimes\mathbb{H}$ on the eight-dimensional $\mathrm{Cl}(3)$ is $\mathrm{Cl}(3)\otimes_{\mathbb{C}}(\mathbb{C}\otimes\mathbb{H})^{\mathrm{op}}\cong M_4(\mathbb{C})\oplus M_4(\mathbb{C})\cong\mathrm{Cl}(5)_{\mathbb{C}}\cong\mathrm{Cl}(6)^+$, of complex dimension $32$ and with a two-dimensional centre, all recomputed exactly on the eight monomials. What remains the source's, and is recorded as such, is the identification of the Coxeter element $\gamma_5$ with a parity-reversal operator.

The Volume Element as the Higgs Candidate

The reading. The source reads the volume element as the Higgs. The argument rests on the one place in the construction where a left-handed state passes to a right-handed one: $\omega=e_1e_2e_3$ is the element that exchanges the two summands, the left-handed electron of the first set and the right-handed electron of the second set are related by multiplication by $\omega$ — checked here, $\omega\,e^-_L=-e^-_R$ and $\omega\,e^-_R=-e^-_L$, with $\omega^2=1$ — and $\omega$ is central and self-adjoint, so it commutes with the whole algebra and carries a global $\mathrm{U}(1)$ phase. In the standard mechanism the Higgs is what supplies the hypercharge shift turning a right-handed fermion into a left-handed one; the source proposes the pseudoscalar $\omega$ of a Clifford algebra as the algebraic object in that role, for $\mathrm{Cl}(3)$ and, with the quarks added, for $\mathrm{Cl}(7)$, where it notes in passing that for the up quark $\omega$ "gets squared to one" and the assignment is therefore not uniform across a generation.

Where the reading stops. The corpus records this as a reading, and names the boundary. What is exact is $\omega$'s centrality, self-adjointness, square and its exchange of the two summands, all recomputed; what is not in the algebra is a coupling. The construction is a finite algebra with no field content, no hypercharge multiplet structure and no Yukawa operator, so "$\omega$ is the Higgs" names an element whose algebraic role resembles the Higgs's role in the standard mechanism, and is not a derivation of a Higgs field, of a Higgs potential, or of a coupling. It is recorded alongside the framework's own scalar-sector ledger rather than as a result of it.

Two claims cited to the standard literature. The source also states, as properties of the left-right symmetric model as such, that the model explains the strong CP problem and that the Higgs coupling vanishes before electroweak symmetry breaking, citing Hall and Harigaya. These are claims about the left-right model and not about the division algebras; the corpus records them as attributed statements of the source's introduction and conclusion, and takes no position on them.

The Sterile Neutrino, Its Mass and the Seesaw

Sterility. The anomaly conditions on a gauge theory with a single $\mathrm{U}(1)$ are $\sum Y=0$, $\sum Y^3=0$ and $\sum YT_3^2=0$ over the fermions, and they constrain only the gauge-charged content. A right-handed neutrino that is a complete gauge singlet carries $Y=0$, $T_3=0$ and colour $0$, so it contributes nothing to any of the three sums; the corpus's anomaly article establishes the same conclusion in the biquaternion framework, that sterile fermions are anomaly-unconstrained. The doubling therefore adds a state that no consistency condition of the gauge sector forbids, and the construction's inclusion of it is not a claim about anomaly cancellation but a claim about the pattern of the fermions.

Mass and the seesaw. A sterile fermion is allowed a Majorana mass $M_R$ as a renormalisable term, unlike the gauge-charged fermions; through the type-I seesaw of The Seesaw Mechanism and Majorana Mass in Biquaternionic Form, a Dirac mass $m_D$ linking $\nu_L$ to $\nu_R$ and a large Majorana mass $M_R$ give a light neutrino of mass $m_\nu\simeq m_D^2/M_R$, so the construction's $\nu_R$ is exactly the state that the seesaw needs and the smallness of the observed neutrino masses follows from the largeness of $M_R$. The corpus's seesaw article works the mechanism in the framework's own conventions: the Dirac mass is the linear chirality-off-diagonal pair, the Majorana block is the self-conjugate pairing on the module real structure, and the renormalisable-level obstruction is the statement that a mass term for a chiral fermion requires a compensating scalar. The construction supplies the second chirality whose presence the seesaw's Dirac mass requires; the mechanism and the scale are the cited standard treatment and are not recomputed here.

Dark matter. A keV-scale sterile neutrino is a standard warm dark matter candidate, produced by oscillation from the active neutrinos in the early universe and detected only gravitationally. The source proposes the construction's right-handed neutrino as this candidate. The corpus records the proposal and its status: the mass scale and the production mechanism are the standard ones of the sterile-neutrino dark-matter literature, the construction's contribution is the presence of the state in the algebra, and the dark-matter identification is a reading in the same sense as the rest.

The proposed probe, and the Majorana requirement. The source names neutrinoless double beta decay as the experiment proposed to infer a right-handed sterile neutrino, and notes that both the seesaw and its own mass-ratio programme require the neutrino to be a Majorana particle. The corpus records both as the standard statements they are — the seesaw needs a Majorana mass for the singlet, and $0\nu\beta\beta$ is the standard lepton-number-violating probe of it — and records that the construction contributes the presence of the state and not the decay rate, the effective mass or any nuclear matrix element. The Neutrino and Majorana Fermions in Biquaternionic Form carries the corpus's treatment of the Majorana condition.

What the Construction Establishes, Reads, and Leaves Open

The construction is best summarised by sorting its steps, as the framework's agendas sort theirs.

Step	Status
$\mathrm{Cl}(3)\cong\mathbb{B}\oplus\mathbb{B}$ and $\mathrm{Cl}(7)\cong\mathrm{Cl}(6)\oplus\mathrm{Cl}(6)$; central volume elements $\omega,J$ with $\omega^2=J^2=1$	Established in the mathematics articles, recomputed
Parity automorphism $\sigma:e_i\mapsto-e_i$; the two inequivalent pinor representations distinguished by the eigenvalue of the volume element	Established in the mathematics articles
The two summands carry the same $\mathrm{su}(3)\oplus\mathrm{u}(1)$ and the same charge spectrum; $[N,J]=[\Lambda_a,J]=0$	Established in the mathematics articles, recomputed
Left and right quaternion actions commute; combined they generate $\operatorname{End}_{\mathbb{C}}(\mathbb{B})\cong M_4(\mathbb{C})$ with $\mathrm{so}(4)=\mathrm{su}(2)\oplus\mathrm{su}(2)$	Established, recomputed on the four-dimensional model
$\omega=e_1e_2e_3$ is central and self-adjoint with $\omega^2=1$, and exchanges the two summands; $\omega\,e^-_L=-e^-_R$	Established, recomputed in $\mathrm{Cl}(0,3)\cong\mathbb{H}\oplus\mathbb{H}$
The triple $\tau_1,\tau_2,\tau_3$ built from the ladder operator $\Omega_L$ closes on $\mathrm{su}(2)$ with $\tau_3=1-2N$; $P=\tfrac12(1+ie_1e_2)$ is a self-adjoint idempotent; $T_a=\tau_a\otimes P$ closes the same way and annihilates $1-P$ identically	Established, recomputed
Ladder operators, primitive idempotent, the eight-dimensional minimal left ideal and the charge spectrum $0,\tfrac13,\tfrac13,\tfrac13,\tfrac23,\tfrac23,\tfrac23,1$ in each summand	Established in the Cl(6) article and inherited by each summand of $\mathrm{Cl}(7)$; recomputed
The right action of $\mathbb{C}\otimes\mathbb{H}$ on $\mathrm{Cl}(3)$ gives $\mathrm{Cl}(5)\cong\mathrm{Cl}(6)^+$; the generated algebra is $M_4(\mathbb{C})\oplus M_4(\mathbb{C})$, of dimension $32$ and centre $2$	The $\mathbb{C}\mathrm{l}(4)\cong M_4(\mathbb{C})$ half was verified here; the $\mathrm{Cl}(5)\cong M_4(\mathbb{C})\oplus M_4(\mathbb{C})$ half is now recomputed in Complex Split Biquaternions and the Clifford Algebra Cl(3); the Coxeter-element parity operator is the source's
The two summands are the two chiralities; the parity automorphism is the discrete left-right symmetry	Reading
The two weak factors act on opposite chiralities, $\mathrm{SU}(2)_L$ on the left-handed and $\mathrm{SU}(2)_R$ on the right-handed states	The selectivity ($T_a(1-P)=0$) is established; which states are in $P$ is a reading
The charge-$0$ state of the right-handed summand is the sterile right-handed neutrino	Reading; the state's gauge quantum numbers are exact, its identification is a reading
The volume element $\omega$ is the Higgs, supplying the hypercharge shift between the two chiralities	Reading; $\omega$'s exchange of the two summands is exact, the Higgs identification and any coupling are not in the algebra
$\Omega_L,\Omega_L^{\dagger}$ ("mimic $W^-$, $W^+$") and $\Omega_R,\Omega_R^{\dagger}$ as their right-handed analogues	Reading of the ladder operators; abandoned for $\mathrm{SU}(2)_R$ in the source's later programme, where $\mathrm{SU}(2)_R$ is used for chiral gravity
One generation of quarks and leptons, both chiralities, in the Pati–Salam multiplets of $\mathrm{SU}(4)_c\times\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$	Reading; the multiplet assignment is not recomputed
$\sqrt{m_e}:\sqrt{m_u}:\sqrt{m_d}=1:2:3$; $\sqrt{m}$ quantised in $(\tfrac13,\tfrac23,1)$ and identified with the second summand's $\mathrm{U}(1)$, read as $\mathrm{U}(1)_{\text{grav}}$	Source's proposal; checked against PDG central values, $1:2.06:3.02$
Three generations from the eigenvalues of $J_3(\mathbb{O})$ with fermionic entries, and from rotations of one generation's representation	Source's proposal, not recomputed
The left-right model explains the strong CP problem and the vanishing of the Higgs coupling before symmetry breaking	Standard-literature claim cited by the source (Hall–Harigaya); not a claim of the algebra
Sterile fermions are unconstrained by the anomaly conditions	Standard, and established in the corpus's anomaly article
Light neutrino mass from the type-I seesaw $m_\nu\simeq m_D^2/M_R$; $M_R$ a renormalisable Majorana mass for the singlet; $0\nu\beta\beta$ as the proposed probe	Standard, treated in the corpus's seesaw article
keV sterile neutrino as warm dark matter	Literature's proposal, recorded
Leptoquark combination of the two chiralities before symmetry breaking, on the doubled $\mathrm{Cl}(7)$, with a merged charge-root-mass	Source's programme, put as a question; the construction supplies no leptoquark field
Three spin-$1$ Lorentz bosons instead of a spin-$2$ graviton; $10$-dimensional octonionic pre-spacetime from $\mathrm{Sl}(2,\mathbb{O})$ on $J_2(\mathbb{O})$, $(1,9)$ signature, $8$-dimensional octonionic Kaluza–Klein	Source's programme, most speculative, recorded not endorsed
The three-colour content, the quantised charge and the observed charge multiplicities	Established in the particle-roster article's sense; the colour and charge successes are properties of a finite algebra

Three statements deserve emphasis.

What the doubling genuinely supplies. The second chirality, as the second simple summand, with the same internal quantum numbers and the same charge spectrum as the first — a property of a finite algebra, not a transcription. This is exactly the structure that the framework's own ledger records as missing: Chiral Fermions in the Biquaternion Framework establishes the projectors and the selection rule $q_L=q_R$ but not a non-central action under which the two chiral halves transform inequivalently, and the doubling supplies such an action by construction, at the price of leaving the biquaternion algebra.
What is a reading. The identification of the two summands with the two chiralities; the identification of the two quaternion actions with the two weak factors; the multiplet assignment; and the identification of the neutral state with the right-handed neutrino. Each is a choice consistent with the exact algebra, not a consequence of it; and, as in the particle-roster article, the same charge multiset can be grouped in more than one way, so the charge table fixes multiplicities and not the grouping.
What remains open. The decisive computations are the same three that the particle-roster article names — the centraliser of $\mathrm{su}(3)\oplus\mathrm{u}(1)$ inside the internal algebra, the projection of the abelian generator on the colour Cartan subalgebra, and its commutator with the Lorentz algebra — together with the choice of the supercharge sector that fixes the isospin alignment of the two quaternion actions. The doubling does not perform them; it supplies the second chirality that makes them meaningful.

The Further Programme of the Source

The construction is presented by its authors as one step of a larger programme, and the corpus records the further steps in the same spirit, as the source's programme rather than as results.

Mass as a square root and a second $\mathrm{U}(1)$. The source proposes that the $\mathrm{U}(1)$ number operator of the right-chiral fermions — the second summand's charge operator, not the hypercharge — be identified with the square root of mass, $\pm\sqrt{m}$, which like electric charge comes with both signs, one sign for matter and the other for antimatter. On that identification the square root of mass is quantised in the ratio $(\tfrac13,\tfrac23,1)$, assigned respectively to the electron, the up quark and the down quark, so that relative to the electric-charge assignment of the first $\mathrm{Cl}(6)$ the positions of the electron and the down quark are interchanged in the second $\mathrm{Cl}(6)$ while the up quark keeps its place. This is what makes the second abelian factor a mass factor rather than a second electric charge, and the source calls it $\mathrm{U}(1)_{\text{grav}}$ and associates it with gravitation; in the pre-theory of the next paragraph the two invariants merge into a single "charge-root-mass", quantised in units $0,\tfrac13,\tfrac23,1$ with both signs. The shift from the earlier sections is worth marking: there the two summands carry the same $\mathrm{U}(1)$ charge, and the reassignment of the second copy's $\mathrm{U}(1)$ from electric charge to square-root mass is a step of the source's later programme, not a property of the doubling. The corpus records the whole chain as a proposal and notes that its own mass articles read mass differently: The Petiau System and the Quantisation of Mass in Biquaternionic Form reads mass from the eigenvalues of $\tilde E\tilde E^{*}$ and from the double-periodic Petiau modulus, and does not use a square root.

Leptoquarks before the breaking. The source puts one further speculation, as a question, at the end of its sixth section: whether before electroweak symmetry breaking the left-handed and the right-handed particles existed in a combined state of leptoquarks, carrying both the electric charge and the square-root-mass charge. In its account the leptoquarks are bosons, described by the doubled algebra $\mathrm{Cl}(7)$ — two copies of $\mathrm{Cl}(6)$ and the split bioctonions — which the source gives as its reason for the study of the split bioctonions. On the merged identification the two quantum numbers become one, the charge-root-mass, quantised in units $0,\tfrac13,\tfrac23,1$ with both signs, the negative sign for antiparticles. The corpus records the proposal as the source's programme and as a question rather than a result: the word leptoquark occurs nowhere else in the corpus, the construction exhibits no leptoquark field, no coupling to the states of the two summands and no decay, and the bosonic character that the source assigns to the combined states is a statement of its pre-theory, not a property of the two Clifford algebras.

The square-root mass ratios and the exceptional Jordan algebra. The $1:2:3$ statement of the source's fourth and sixth sections is about the square roots of the masses of the electron, the up quark and the down quark — one generation, leptons and quarks together, not the charged leptons alone — and is quoted as the motivation for the mass-ratio programme rather than derived in this paper. Checked here arithmetically against the PDG central values $m_e=0.5110$ MeV, $m_u=2.16$ MeV, $m_d=4.67$ MeV, the ratio of square roots is $1:2.06:3.02$, which reproduces the pattern to a few per cent (the comparison is scheme- and scale-dependent, since the light-quark masses are). The separate three-generation statement is that the eigenvalues of the exceptional Jordan matrices $J_3(\mathbb{O})$ with fermionic entries give the mass ratios of the three Standard Model generations, summarised in the source's Figure 2 and worked out in its own reference on the fine-structure constant and mass ratios; the source proposes obtaining the three generations themselves as rotations of the fermionic representation of any one generation, a transformation of $J_3(\mathbb{O})$. Octonions and the Exceptional Lie Groups treats $J_3(\mathbb{O})$ as the carrier of the exceptional structures, and the corpus records both statements as the source's proposals: it does not recompute the eigenvalue fit, the fermionic-entry construction or the three-generation rotation, and it corrects here an earlier corpus sentence that had called the $1:2:3$ pattern a charged-lepton ratio and had attributed it directly to the $J_3(\mathbb{O})$ eigenvalues, which are two different claims of the source.

$\mathrm{E}_6$, $\mathrm{Spin}(10)$ and the exceptional subgroups. The complexified exceptional group $\mathrm{E}_6$ is the complexification of $\mathrm{F}_4$, and the source uses the maximal subgroups of $\mathrm{F}_4$ — the two isomorphic subgroups $H_1=[\mathrm{SU}(3)\times\mathrm{SU}(3)]/\mathbb{Z}_3$ and $H_2=\mathrm{Spin}(9)$, whose intersection $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ is the Standard Model's gauge group — and their complexified counterparts in $\mathrm{E}_6$: the subgroups $\tilde H_1=[\mathrm{SU}(3)]^3/\mathbb{Z}_3$ and $\tilde H_2=\mathrm{Spin}(10)$, whose intersection $$ \mathrm{SU}(3)\times\mathrm{SU}(2)_R\times\mathrm{SU}(2)_L\times\mathrm{U}(1) $$ is the left-right symmetric gauge group. These subgroup statements are standard Lie theory, cited by the source from the literature of Boyle and of Dubois-Violette and Todorov; the corpus records them as standard and not as results of the construction.

$\mathrm{E}_8\times\mathrm{E}_8$, trace dynamics and pre-spacetime. The source's further programme proposes an $\mathrm{E}_8\times\mathrm{E}_8$ structure, one factor for each chirality, as the carrier of a pre-gravitational interaction; a passage from that structure to quantum theory and gravity through Adler's trace dynamics, in which quantum mechanics emerges as the statistical mechanics of a deterministic matrix mechanics; and a pre-spacetime in which the algebra precedes and generates the spacetime manifold. Two consequences are stated sharply enough to record separately. The first is what happens to $\mathrm{SU}(2)_R$: rather than being the gauge factor of a right-handed weak interaction, it is used for a chiral formulation of gravity, and in place of a spin-$2$ graviton the programme proposes three spin-$1$ Lorentz bosons, on an equal footing with the other gauge bosons — a departure from the left-right model as usually written, which the corpus marks. The second is the arena: before symmetry breaking the spacetime is taken to be ten-dimensional and labelled by the octonions, its symmetry being the action of $\mathrm{Sl}(2,\mathbb{O})$ on the Hermitian $2\times2$ octonionic matrices $J_2(\mathbb{O})$, whose determinant gives a $(1,9)$ signature, so that the Lorentz–weak symmetry is unified with the electro-colour symmetry through an eight-dimensional octonionic Kaluza–Klein structure; the $J_3(\mathbb{O})\supset J_2(\mathbb{O})\supset J_2(\mathbb{C})$ stabiliser chain of the preceding paragraph is read as the chain in which the stabiliser of $4$-dimensional spacetime sits inside the stabiliser of the $10$-dimensional one, giving the Standard Model group at the bottom. These are the programme's most speculative steps. The corpus records them as such: they are not derived from the two Clifford algebras of this article, and the corpus's ledger already records that the framework's ceiling is the reason such structures have to be supplied outside the biquaternion algebra. The Coleman–Mandula theorem, which the source invokes, forbids the combination of spacetime and internal symmetries of the ordinary kind; the programme argues that in a pre-spacetime with no background manifold the distinction between internal and spacetime symmetries has not yet arisen, so the theorem does not apply, and the corpus records the argument without endorsing it.

Summary

The left-right symmetric construction of Vaibhav and Singh rests on two exact algebraic facts and a set of readings. The exact facts are the doublings $\mathrm{Cl}(3)\cong\mathrm{Cl}(2)\oplus\mathrm{Cl}(2)$ and $\mathrm{Cl}(7)\cong\mathrm{Cl}(6)\oplus\mathrm{Cl}(6)$, in which the central volume element — an involution, $\omega^2=J^2=1$ — supplies a complete pair of central idempotents and hence two summands, the two inequivalent pinor representations of the algebra, interchanged by the parity automorphism $e_i\mapsto-e_i$; and the two commuting quaternion actions of the biquaternion algebra, whose combination generates $\operatorname{End}_{\mathbb{C}}(\mathbb{B})\cong M_4(\mathbb{C})$ with the $\mathrm{su}(2)\oplus\mathrm{su}(2)$ of $\mathrm{so}(4)$. The readings are the identification of the two summands with the two chiralities and of the two quaternion actions with $\mathrm{SU}(2)_L$ and $\mathrm{SU}(2)_R$, the identification of the neutral state of the right-handed summand with the sterile right-handed neutrino, the assignment of the states to the Pati–Salam multiplets, and the reading of the volume element $\omega$ as the Higgs. The construction's contact with the framework's ledger is that the second chirality and the sterile state are supplied by the algebra rather than added by hand, and that a sterile fermion is unconstrained by the anomaly conditions; its open end is the same as that of the undoubled construction, namely the three commutator computations that would decide whether the abelian charge is a physical gauge charge, together with the choice of the supercharge sector that fixes the isospin alignment. Further parts of the source's fifth and sixth sections were added to this record on later passes and are worth naming: the explicit construction of the isospin generators from the ladder operator, whose selectivity — the generators annihilating the complementary component of the algebra identically — was recomputed here; the source's proposal that the two chiralities existed in a combined leptoquark state before symmetry breaking; the source's later programme, in which $\mathrm{SU}(2)_R$ is reassigned from a weak factor to a chiral formulation of gravity with three spin-$1$ Lorentz bosons, the spacetime becomes a ten-dimensional octonionic pre-spacetime, and the second summand's $\mathrm{U}(1)$ is read as the square root of mass; and the recomputation, in the companion mathematics article, of the $\mathrm{Cl}(5)$ statement that the right action of $\mathbb{C}\otimes\mathbb{H}$ on $\mathrm{Cl}(3)$ generates $M_4(\mathbb{C})\oplus M_4(\mathbb{C})\cong\mathrm{Cl}(5)_{\mathbb{C}}$, previously left as the source's. Like the particle-roster construction, it is outside the biquaternion algebra, and it is recorded here for the same reason: it is the literature's most economical answer to the chirality and sterile-neutrino questions, and its successes and its readings are best kept apart.

Summary of Notation

Symbol	Meaning
$\mathrm{Cl}(3)\cong\mathbb{B}\oplus\mathbb{B}$	Complex split biquaternions; two biquaternion summands
$\mathrm{Cl}(7)\cong\mathrm{Cl}(6)\oplus\mathrm{Cl}(6)$	Complex split bioctonions; two octonionic chain-algebra summands
$\omega=E_1E_2E_3$, $J=e_1\cdots e_7$	Central volume elements, $\omega^2=J^2=1$
$\Pi_\pm=\tfrac12(1\pm\omega),\ \tfrac12(1\pm J)$	Central idempotents projecting on the two summands
$\sigma:e_i\mapsto-e_i$	Parity automorphism; the discrete left-right symmetry
$L_x,R_x$	Left and right quaternion multiplications; commuting $\mathrm{su}(2)$'s
$M_4(\mathbb{C})\cong\mathbb{C}\mathrm{l}_4$	Algebra generated by the two actions; contains $\mathrm{so}(4)\cong\mathrm{su}(2)\oplus\mathrm{su}(2)$
$\mathrm{Cl}(3)\otimes_{\mathbb{C}}(\mathbb{C}\otimes\mathbb{H})^{\mathrm{op}}\cong M_4(\mathbb{C})\oplus M_4(\mathbb{C})\cong\mathrm{Cl}(5)_{\mathbb{C}}\cong\mathrm{Cl}(6)^+$	Algebra generated by left $\mathrm{Cl}(3)$ and right $\mathbb{C}\otimes\mathbb{H}$; dimension $32$, centre $2$
$\Omega_L=\alpha_1\alpha_2\alpha_3$, $\Omega_R$	Composite ladder operators of the two summands; read as $W^\pm$
$\tau_1=\Omega_L+\Omega_L^{\dagger}$, $\tau_2=i(\Omega_L-\Omega_L^{\dagger})$, $\tau_3=\Omega_L\Omega_L^{\dagger}-\Omega_L^{\dagger}\Omega_L$	$\mathrm{su}(2)$ triple from the ladder operator, $\tau_3=1-2N$
$T_a=\tau_a\tfrac12(1+ie_1e_2)$	Isospin generators of the doubled ideal; annihilate the complementary component
$\beta_1=\tfrac12(-e_1+ie_2\tau_3)$, $\beta_2=\Omega_L^{\dagger}ie_2$	Totally isotropic elements of the right action
$V_R,V_L,e^-_R,e^-_L$	The four states of the right-action ideal
$\mathrm{SU}(2)_L,\mathrm{SU}(2)_R$	Weak factors read from the left and right actions (reading)
$\mathrm{SU}(4)_c\times\mathrm{SU}(2)_L\times\mathrm{SU}(2)_R$	Pati–Salam gauge group of the reading
$\nu_R$	Right-handed neutrino; $\mathrm{SU}(2)_L$ singlet, sterile (reading)
Leptoquark	Source's proposed pre-symmetry-breaking combined state of the two chiralities, on the doubled $\mathrm{Cl}(7)$
$m_\nu\simeq m_D^2/M_R$	Type-I seesaw mass; $M_R$ the singlet Majorana mass
$\sqrt{m}$	Square root of mass; the second summand's $\mathrm{U}(1)$, quantised $(\tfrac13,\tfrac23,1)$
$J_3(\mathbb{O})$	Exceptional Jordan algebra; source's carrier of the three-generation mass ratios
$\mathrm{E}_6\cong\mathrm{F}_4\otimes\mathbb{C}$	Complexified exceptional group of the programme
$\mathrm{U}(1)_{\text{grav}}$	Proposed second abelian factor of the programme
$J_2(\mathbb{O})$, $\mathrm{Sl}(2,\mathbb{O})$	Pre-spacetime arena of the programme; determinant of $(1,9)$ signature