Squeezing Beyond the Biquaternion Algebra

Introduction

Squeezing is the reduction of a state's uncertainty in one variable at the expense of its conjugate. For a spin-0 oscillator it is generated by a quadratic function of the creation and annihilation operators, and in the biquaternion framework those operators are central: the squeeze operator is a central unitary element, it acts on the envelope and leaves the state module untouched. Squeezing, in that sense, is inside the algebra, and the preceding article's oscillator is its natural home. The purpose of this article is to determine exactly where squeezing stops being expressible inside $\mathbb{B}$, and what lies beyond.

Three boundaries are examined, and each is a different kind of "beyond".

Beyond the spin-0 reach. A squeeze operator built from central operators can squeeze the quadratures of the envelope, but it cannot squeeze the module, because a central unitary multiplies both components of a state-module element by the same phase and does not change their relative weights. Squeezing the module requires a non-central generator, which is spin squeezing and belongs to the sibling spin subcategories. This is the boundary of the present subcategory.
Beyond the algebra by contraction. A contraction is a singular limit that degenerates the algebra: generators are rescaled so that some commutators and some norms go to zero. The quaternion algebra contracts to the algebra of complex numbers with nilpotents, the biquaternion norm degenerates, and elements that were invertible become zero divisors. The zero divisor cone is the limiting locus of that degeneration. This is a boundary of the algebra's quadratic structure.
Beyond the algebra by deformation, enlargement, or completion. q-deformations replace the commutators by q-commutators and turn the algebra into a Hopf algebra with no invariant quadratic form of the same kind; Clifford enlargements, octonionic doublings, and the infinite-dimensional or non-unitary algebras of field theory and open systems each lie outside $\mathbb{B}$ for a definite algebraic reason. This is the boundary of associativity, finite dimensionality, and unitarity.

The article states each boundary, computes what can be computed, and marks the rest. The biquaternion framework's contribution is to make the boundaries precise: it says which manipulations of the oscillator are central and therefore available, and which require an operation the algebra does not contain.

The article is organised as follows. The next section sets up squeezing inside the algebra and verifies the squeezed-vacuum uncertainty product. The third section explains why the module cannot be squeezed centrally and where spin squeezing lives. The fourth treats contractions and the degeneration of the biquaternion norm. The fifth treats q-deformations, Clifford and octonionic enlargements, and infinite-dimensional or non-unitary extensions. The sixth collects the boundary statements, and the closing sections are the summary, the notation table, and the external literature.

The conventions are those of the companion articles: $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ with $e_0=1,e_1,e_2,e_3$, $e_j^2=-e_0$, central $i$; $\mathbb{M}_\pm$ are the Hermitian and anti-Hermitian sectors; $\mathbb{C}_{\mathbb{B}}$ is the center; $\mathbb{H}_{\mathbb{B}}$ is the real-quaternion subspace; the state module is $\mathbb{B}\tilde\Pi\cong\mathbb{C}^2$; $\mathrm{Tr}(\tilde H)=2\,\mathrm{Sc}(\tilde H)$; and the biquaternion norm is $N(x)=x\bar x$.

Squeezing Inside the Algebra

The squeeze operator and its centrality

For a one-dimensional oscillator with mass $m$ and frequency $\omega$, the creation and annihilation operators are

$$ a=\sqrt{\frac{m\omega}{2\hbar}}\,\hat x+\frac{i}{\sqrt{2m\hbar\omega}}\,\hat p, \qquad a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}\,\hat x-\frac{i}{\sqrt{2m\hbar\omega}}\,\hat p, \qquad [a,a^\dagger]=e_0 . $$

Both $\hat x$ and $\hat p=-i\hbar\partial_x$ are central operators: $\hat x$ multiplies by a real scalar and $\hat p$ is the central imaginary times a real derivative. Hence $a$ and $a^\dagger$ are central, and the squeeze operator

$$ S(\xi)=\exp\left[\tfrac12\left(\bar\xi\,a^2-\xi\,a^{\dagger 2}\right)\right], \qquad \xi=re^{\,i\theta}, $$

is a central element of $\mathbb{B}$. Its generator $\tfrac12(\bar\xi a^2-\xi a^{\dagger2})$ is anti-Hermitian, so $S$ is unitary in the framework's sense,

$$ S\,S^{*}=e_0 , $$

and it is a unitary central element, not a unit-norm rotor. It acts on a state-module element by scalar multiplication,

$$ \psi\ \longmapsto\ S\,\psi = S\otimes I_2\;\psi , $$

so it changes the envelope and leaves the module orientation exactly as it was. The squeeze operator does not squeeze the module.

The standard Bogoliubov transformation follows from the central algebra,

$$ S^{*}\,a\,S=a\cosh r-e^{\,i\theta}\sinh r\,a^\dagger , \qquad S^{*}\,a^\dagger\,S=a^\dagger\cosh r-e^{-i\theta}\sinh r\,a , $$

and in terms of quadratures it is the general Bogoliubov dilation. Writing $\hat x$ and $\hat p$ in terms of $a$ and $a^\dagger$ and using their transformation,

$$ S^{*}\hat xS=\sqrt{\frac{\hbar}{2m\omega}}\Big[(a+a^\dagger)\cosh r-\sinh r\big(e^{\,i\theta}a^\dagger+e^{-i\theta}a\big)\Big], $$

which for real $\xi$ ($\theta=0$) reduces to the pure dilation $S^{*}\hat xS=e^{-r}\hat x$ and $S^{*}\hat pS=e^{r}\hat p$, while for $\theta=\pi/2$ it becomes a rotation–dilation mixing $\hat x$ and $\hat p$. The transformation is a central Bogoliubov rotation; in the biquaternion reading it is a central automorphism of the oscillator algebra, and it never touches the module. The transformation is standard; the framework's statement is that its generator is central.

The squeezed vacuum and the uncertainty product

The squeezed vacuum is $S(\xi)|0\rangle$, a Gaussian whose width in $x$ is reduced by $e^{-r}$. In the position representation it is

$$ \psi_r(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}e^{\,r/2} \exp\left[-\frac{m\omega}{2\hbar}\,e^{2r}\,x^2\right], $$

up to a phase. The quadrature variances are

$$ \Delta x^2=\frac{\hbar}{2m\omega}e^{-2r}, \qquad \Delta p^2=\frac{\hbar m\omega}{2}e^{2r}, \qquad \Delta x\,\Delta p=\frac{\hbar}{2}, $$

so the state is a minimum-uncertainty state at every $r$: squeezing redistributes the uncertainty between the quadratures but cannot reduce the product below $\hbar/2$. This is the algebraic reason squeezing is a legitimate operation of the framework: it is a central automorphism of the oscillator algebra, and the uncertainty product is fixed by the central commutator $[\hat x,\hat p]=i\hbar e_0$.

The variances were verified numerically by direct quadrature in the position representation, in explicit arithmetic with $m=\hbar=\omega=1$ and the width convention $\alpha=(m\omega/\hbar)e^{2r}$. The squeezed vacuum is not a single oscillator state but a superposition of all the even-numbered states, and the computation used its position wave function in that superposition:

$$ \begin{aligned} r=0.5:&\quad \Delta x^2=0.18393972=\tfrac{\hbar}{2m\omega}e^{-1},\quad \Delta p^2=1.35914091=\tfrac{\hbar m\omega}{2}e^{1},\quad \Delta x\Delta p=0.50000000,\\[2pt] r=0.7:&\quad \Delta x^2=0.12329848=\tfrac{\hbar}{2m\omega}e^{-1.4},\quad \Delta p^2=2.02759998=\tfrac{\hbar m\omega}{2}e^{1.4},\quad \Delta x\Delta p=0.50000000 . \end{aligned} $$

Both rows reproduce the closed forms to eight decimals, and the product is $\hbar/2$ in both cases.

The biquaternion reading of the squeeze

The whole squeezing construction lives in the center. The generator is $\tfrac12(\bar\xi a^2-\xi a^{\dagger2})$, a central anti-Hermitian element; the squeeze operator is a central unitary; the quadrature dilation is a central automorphism; the squeezed vacuum's Gaussian envelope is a central real function; the uncertainty product is fixed by a central commutator. In the framework's vocabulary, squeezing is an operation of the classical sector on the envelope: it is a symmetry of the central oscillator algebra that changes the shape of the state without changing its module content. This is why squeezing appears in the spin-0 subcategory at all, and it is the sense in which it is "inside" the algebra.

Composition and two-mode squeezing

Squeezes compose by adding their rapidities in the same direction: $S(r_1\hat n)S(r_2\hat n)=S((r_1+r_2)\hat n)$ for a fixed squeezing axis $\hat n$, because the one-parameter subgroups of the corresponding central generators add. The composition was checked at the level of the variances: squeezing twice by $r_1=0.3$ and $r_2=0.4$ gives the same width $e^{-(r_1+r_2)}=e^{-0.7}$ as a single squeeze of $r=0.7$, which is the width already tabulated. Squeezes along different axes do not commute, because their generators do not: the resulting group is the metaplectic group acting on the envelope, and its whole action is central.

A two-mode version is also inside the algebra and is worth recording because it is the strongest central correlation available. For two oscillator modes $a_1,a_2$ the two-mode squeeze operator

$$ S_{12}(\xi)=\exp\left[\bar\xi\,a_1a_2-\xi\,a_1^\dagger a_2^\dagger\right] $$

has a central generator and is a central unitary, and it creates correlations between the two modes — the two-mode squeezed vacuum is the standard entangled state of quantum optics. In the framework the statement is that the algebra supports central correlations between modes (a correlation between two envelopes, each a scalar function), but not correlations inside a single spin-0 module, which is the boundary of the next section. The two-mode operator is central; the operation that would entangle the two components of one module is not.

Squeezing the Module: Beyond the Spin-0 Reach

A different operation suggests itself: squeeze the module. The state module $\mathbb{B}\tilde\Pi\cong\mathbb{C}^2$ carries the two components of a state, and one could ask for a transformation that reduces the uncertainty of one component at the expense of the other. Such a transformation is not central. A central unitary multiplies both components by the same phase, so it acts on the module as the identity matrix $I_2$ and cannot change the relative weights or the correlations of the components. To redistribute uncertainty inside the module one needs an operator whose module action is a non-trivial element of $U(2)$ — a rotation generated by the $e_k$, the non-central generators.

In the framework those generators are exactly the ones that produce the spin structure: $\Phi(ie_k)=-i\sigma_k$, and a rotation $e^{i\alpha e_k}$ acts on the module as a spin rotation. A squeeze built from such a generator is a spin-squeezing operation, and it uses precisely the non-central part of the algebra that the spin-0 subcategory excludes. The framework's statement is sharp: the operator that squeezes the module is a rotation of the module, and the sibling subcategories treat it. The companion article Angular Momentum and Spin in Biquaternionic Form develops the rotation generators; the spin-squeezing problem itself is the subject of a spin subcategory article.

It is worth being precise about what is and is not claimed. The central squeeze cannot change the module orientation; it can still change the module's magnitude relative to the envelope (a global scale), and it can change the correlations between the envelope's quadratures and a fixed module basis. But a genuine reduction of one module component's uncertainty at the cost of the other — the operation a spin-squeezing experiment performs — requires a non-central invariant, and the framework provides it only in the spin subcategories. In the spin-0 subcategory the module is a spectator by construction.

The contrast is visible in the module's matrix image. Under $\Phi$, a central unitary is $e^{i\phi}I_2$ and a non-central rotation generated by $e_1$ is $e^{\,i\alpha\sigma_1/2}$. Acting on the module element $(1,0)^T$,

$$ e^{\,i\phi}I_2\begin{pmatrix}1\\0\end{pmatrix}=e^{\,i\phi}\begin{pmatrix}1\\0\end{pmatrix}, \qquad e^{\,i\alpha\sigma_1/2}\begin{pmatrix}1\\0\end{pmatrix} =\begin{pmatrix}\cos(\alpha/2)\\ i\sin(\alpha/2)\end{pmatrix}, $$

so the central operation preserves the component weights while the non-central one transfers weight from the first component to the second. The second is a module rotation; a squeeze built on it is spin squeezing. The identity of the matrix images and the transfer of weight were checked in explicit two-by-two arithmetic. The central squeeze, by contrast, cannot write a nonzero second component into a state whose second component is zero: it is a scalar, and scalars do not mix components.

Contracting the Algebra

Contraction and the degeneration of the biquaternion norm

A contraction of a Lie algebra rescales a subset of the generators so that some structure constants tend to zero, producing a different, typically non-semisimple algebra in the limit. For $\mathbb{B}$ the standard route is a rescaled quaternion basis. Let $\epsilon>0$ and set

$$ E_1=e_1, \qquad E_2=\epsilon\,e_2, \qquad E_3=\epsilon\,e_3 . $$

Then

$$ E_1^2=-e_0, \qquad E_2^2=E_3^2=-\epsilon^2 e_0, \qquad [E_2,E_3]=2\epsilon^2 E_1 , \qquad [E_1,E_2]=2E_3,\quad [E_1,E_3]=-2E_2 , $$

so that as $\epsilon\to0$ the commutators involving $E_2,E_3$ vanish and the algebra contracts to the direct sum of the complex numbers generated by $E_1$ and a two-dimensional abelian nilpotent piece. The rescaling was verified numerically: the commutator $[E_2,E_3]=2\epsilon^2E_1$ and the biquaternion norm $N(E_2)=\epsilon^2e_0$ both scale as $\epsilon^2$ and vanish together, at $\epsilon=1,0.1,0.01$ giving the values $2,2\times10^{-2},2\times10^{-4}$ and $1,10^{-2},10^{-4}$ respectively.

The essential point is that the biquaternion norm degenerates with the algebra. For $\epsilon>0$ the biquaternion norm on the real span of $E_1,E_2,E_3$ is positive definite and every nonzero element of that real span is invertible; at $\epsilon=0$ the elements $E_2,E_3$ have $N=0$ despite being nonzero, and are nilpotent zero divisors. The contraction takes a real basis with a nondegenerate biquaternion norm to one with a degenerate biquaternion norm. This is the algebraic mechanism behind the zero divisor locus: zero divisors are what survive when a quadratic structure is squeezed until its metric collapses. The companion article Zero Divisors as a Physical Locus in Biquaternionic Form treats the locus itself; here it appears as the endpoint of a contraction.

Contractions to split and degenerate algebras

Other contractions produce the neighbouring algebras. Contracting in the timelike direction produces a split (indefinite) structure in which some generators square to $+e_0$; the resulting algebra has zero divisors built from the lightlike combinations. In an explicit matrix representation with generators $I,J,K$ obeying

$$ I^2=-1, \qquad J^2=+1, \qquad IJ=K , $$

the elements $1+J$ and $1+K$ are zero divisors:

$$ (1+J)(1-J)=0, \qquad (1+K)(1-K)=0 . $$

Both identities were verified in the two-dimensional representation $I=i\sigma_z$, $J=\sigma_x$: the products vanish exactly, while $I^2=-1$ and $J^2=+1$ were confirmed. The split algebra is what one obtains when the biquaternion norm is contracted in one direction, and its zero divisors are the real image of the biquaternion null cone. The real quaternion algebra $\mathbb{H}$ is a division algebra — every nonzero element of $\mathbb{H}$ has $N(x)>0$ and is invertible — but $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ is not: the complex coefficients supply the null cone that the last article of this subcategory treats. The contraction exhibits that locus in a real basis, where it becomes the lightlike cone of the split algebra.

Deforming and Enlarging

q-deformations

A deformation replaces the commutators by q-commutators. The standard example is the q-deformed angular momentum algebra, with

$$ [J_3,J_\pm]=\pm J_\pm, \qquad [J_+,J_-]=\frac{q^{\,2J_3}-q^{-2J_3}}{q-q^{-1}}, $$

which reduces to the undeformed $\mathrm{SU}(2)$ relations as $q\to1$. The limit was verified numerically: at $J_3=1$ the deformed bracket takes the values $2.0333$, $2.0001$, and $2.000001$ for $q=1.2,1.01,1.001$, approaching the undeformed value $2J_3=2$.

The q-deformed algebra lies beyond the biquaternion algebra in a definite sense. It is a Hopf algebra, not simply a unital associative algebra; its representation theory is not the representation theory of $\mathbb{H}$ or $\mathbb{B}$; and, crucially for this corpus, it does not carry an invariant positive-definite quadratic form playing the role of $N$. The biquaternion norm is the framework's central object — it defines unitarity, the sectors, and the zero divisor cone — and a q-deformation does not preserve it. Whether a q-deformed biquaternion structure with an appropriate q-biquaternion norm exists, and what its physical reading would be, is not addressed here; the framework's conventions do not contain such an object, and inventing one would go beyond the algebra the corpus uses.

Clifford enlargements and octonionic doubling

The biquaternion algebra sits inside the Clifford family: $\mathbb{H}\cong\mathrm{Cl}_{0,2}$ and $\mathrm{Cl}_{0,3}\cong\mathbb{H}\oplus\mathbb{H}$, and the Dirac article of the series uses the Clifford algebra of the spacetime metric. Enlarging to $\mathrm{Cl}_{p,q}$ with more generators, or taking the complexification as a Clifford algebra in more dimensions, produces algebras of matrices over $\mathbb{R},\mathbb{C},\mathbb{H}$ of larger size, with no finite-dimensional faithful module of the same two-component kind. Such enlargements lie beyond the algebra in the sense that the state module, the two sectors, and the trace pairing all change dimension and structure.

The Cayley–Dickson doubling gives a sharper boundary. Doubling $\mathbb{C}$ gives $\mathbb{H}$, and doubling $\mathbb{H}$ gives the octonions $\mathbb{O}$, which are non-associative:

$$ (xy)z\ne x(yz) \qquad \text{for some } x,y,z\in\mathbb{O}. $$

The biquaternion framework uses associativity at every step — the module action, the trace, the conjugation identities, and the factorisation $\tilde H=h_0e_0$ of a central operator all presuppose it. A non-associative enlargement therefore lies beyond the algebra not by a matter of convenience but by a structural obstruction: the trace pairing and the module action are not defined without associativity. The octonions are the first Cayley–Dickson algebra to fail this, and they mark the boundary.

Infinite-dimensional and non-unitary extensions

Two further extensions are outside $\mathbb{B}$ for different reasons. The infinite-dimensional algebras of quantum field theory — the canonical commutation relations of a field, the Virasoro and Kac–Moody algebras, the algebra of local observables — have no finite trace and no finite-dimensional module, so the framework's trace pairing $\mathrm{Tr}(\tilde H)=2\,\mathrm{Sc}(\tilde H)$ has no direct analogue. The non-unitary (dissipative) algebras of open systems, generated by Lindblad operators, take the evolution out of the unitary group: the semigroup $e^{\mathcal{L}t}$ is not unitary, a state's purity is not conserved, and the biquaternion norm's value changes with time. The informational reading of such evolution belongs to the informational subcategory.

The framework's central statement about unitarity — that a central Hamiltonian generates a unitary flow, $e^{-i\tilde Ht/\hbar}$ with $\tilde H$ Hermitian and central — is exactly what fails in each of these cases: a field-theoretic Hamiltonian has no finite trace, and a Lindbladian is not anti-Hermitian. Both are honest extensions, and both are beyond the algebra as the corpus uses it.

What the Algebra Cannot Reach

The three boundaries can be stated compactly.

Operation	Inside $\mathbb{B}$?	Why
Envelope squeezing (quadratures)	Yes	Central generator, central unitary, module spectator
Module squeezing (spin squeezing)	No	Requires a non-central generator; sibling spin subcategories
Contraction (degenerating the biquaternion norm)	Limiting case	The limit is a different, degenerate algebra with zero divisors
q-deformation	No	No invariant quadratic form of the framework's kind
Clifford enlargement	No	The module and trace pairing change dimension and structure
Octonionic doubling	No	Non-associativity destroys the module action and the trace
Field-theoretic / dissipative algebra	No	No finite trace; non-unitary evolution

The table is the article's answer: envelope squeezing is available, module squeezing is not, and every operation that deforms, contracts, enlarges, or completes the algebra takes it outside the class where the biquaternion norm, the sectors, and the trace pairing are defined.

Open Questions

1. Is there a q-deformed biquaternion norm? The framework's structure is carried by the quadratic form $N$. Whether a q-analogue exists that reduces to $N$ as $q\to1$ and supports a sector decomposition is unknown; if it does, the q-deformed theory would have a biquaternion reading, and if it does not, the deformation is a genuine boundary.

2. Can a contraction be given a physical reading? The contraction to the split algebra produces the light cone zero divisors; whether the contraction parameter has a physical interpretation — a decoherence rate, an infrared cutoff, a strong-field limit — is not established here.

3. What is the octonionic obstruction for the module? The failure is associativity, which the trace and module action require. Whether a weakened (alternative) structure could carry a two-component module with a trace-like pairing is not explored.

4. Empirical content. Squeezing is experimentally realized for spin-0 light and oscillator modes; the reformulation's content here is organisational rather than predictive, and no prediction distinguishing it from standard squeezing theory is offered.

Summary

Squeezing is generated by $S(\xi)=\exp[\frac12(\bar\xi a^2-\xi a^{\dagger2})]$, whose generator is built from the central operators $a,a^\dagger$ and is therefore central. $S$ is a central unitary, $SS^\dagger=e_0$, and it acts on the state module as $S\otimes I_2$: it squeezes the envelope and leaves the module orientation untouched. The squeezed vacuum $\psi_r\propto e^{-(m\omega/2\hbar)e^{2r}x^2}$ has variances $\Delta x^2=\frac{\hbar}{2m\omega}e^{-2r}$, $\Delta p^2=\frac{\hbar m\omega}{2}e^{2r}$, $\Delta x\Delta p=\hbar/2$, verified by direct quadrature at $r=0.5$ and $r=0.7$ to eight decimals with the product equal to $\hbar/2$.

Squeezing the module is beyond the spin-0 reach: a central unitary cannot change the relative weights of the two module components, and a module squeeze requires a non-central generator, which is a spin rotation and belongs to the sibling spin subcategories.

Contracting the algebra by $E_2=\epsilon e_2$, $E_3=\epsilon e_3$ degenerates the biquaternion norm: $[E_2,E_3]=2\epsilon^2E_1$ and $N(E_2)=\epsilon^2e_0$ vanish together as $\epsilon\to0$ (verified at $\epsilon=1,0.1,0.01$), and the limit contains nilpotent zero divisors. In the split algebra with $I^2=-1$, $J^2=+1$, the elements $1+J$ and $1+K$ are zero divisors, $(1+J)(1-J)=(1+K)(1-K)=0$, verified exactly in the representation $I=i\sigma_z$, $J=\sigma_x$.

q-deformations, with $[J_+,J_-]=\frac{q^{2J_3}-q^{-2J_3}}{q-q^{-1}}\to2J_3$ as $q\to1$ (verified at $q=1.2,1.01,1.001$), lie beyond the algebra because no invariant quadratic form of the framework's kind survives. Clifford enlargements change the module and trace pairing; the octonionic doubling is non-associative and therefore destroys the module action and the trace; and the field-theoretic and dissipative algebras have no finite trace or no unitary evolution. Envelope squeezing is inside the algebra; module squeezing, contractions, deformations, enlargements, and completions are beyond it, each for a definite structural reason.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra
$e_0=1,e_1,e_2,e_3$	Quaternion basis, $e_j^2=-e_0$
$i$	Central scalar imaginary
$\mathbb{C}_{\mathbb{B}}$	Center; home of the squeeze operator
$\mathbb{H}_{\mathbb{B}}$	Real-quaternion subspace
$\mathbb{M}_+$, $\mathbb{M}_-$	Hermitian and anti-Hermitian sectors
$\mathbb{B}\tilde\Pi\cong\mathbb{C}^2$	State module
$N(x)=x\bar x$	Biquaternion norm
$a,a^\dagger$	Central oscillator operators, $[a,a^\dagger]=e_0$
$S(\xi)=\exp[\frac12(\bar\xi a^2-\xi a^{\dagger2})]$	Squeeze operator; central unitary
$\xi=re^{i\theta}$	Squeeze parameter
$S^{*} aS=a\cosh r-e^{i\theta}\sinh r\,a^\dagger$	Bogoliubov transformation
$\Delta x^2=\frac{\hbar}{2m\omega}e^{-2r}$, $\Delta p^2=\frac{\hbar m\omega}{2}e^{2r}$	Squeezed-vacuum variances
$E_k=\epsilon e_k$	Contracting basis
$\mathrm{Cl}_{p,q}$, $\mathbb{O}$	Clifford algebras; octonions (non-associative)
$[J_+,J_-]=\frac{q^{2J_3}-q^{-2J_3}}{q-q^{-1}}$	q-deformed bracket
$\mathrm{Tr}(\tilde H)=2\,\mathrm{Sc}(\tilde H)$	Trace convention