The Conformal Group in Biquaternionic Form

Introduction

The companion articles have established the homogeneous Lorentz group inside the biquaternion algebra exactly. The unit-norm biquaternions are $SL(2,\mathbb{C})$, the double cover of the restricted Lorentz group $SO^+(1,3)$, and rotor conjugation $$ \tilde{Q}\;\longmapsto\;\tilde{\Lambda}\,\tilde{Q}\,\tilde{\Lambda}^{*}, \qquad \tilde{Q}\in\mathbb{M}_-,\quad \tilde{\Lambda}\tilde{\Lambda}^{\natural}=e_0, $$ is their action on the material sector $\mathbb{M}_-$. The Poincaré article then showed that the translations are of a different kind: a shift is affine, not linear, so no rotor generates it, and the restricted Poincaré group is carried as pairs $(\tilde{\Lambda},\tilde{a})$ with $\tilde{a}\in\mathbb{M}_-$, a semidirect product in which the algebra supplies the homogeneous factor and its action on the shift but not the shift itself. That article closed with an explicit open question — whether an enlarged algebra could absorb translations multiplicatively, and whether a conformal extension could realize them "as parabolic products of generalized inversions rather than as pure rotors."

This article takes up the conformal end of that question. The conformal group of Minkowski space is the fifteen-parameter group generated by the Poincaré group together with the dilations and the special conformal transformations; it is $SO(2,4)$, double-covered by $SU(2,2)$. The question is how much of it the biquaternion algebra $\mathbb{B}$ carries.

The answer is a boundary rather than an embedding, and the boundary falls in a definite place. The algebra carries the conformal structure: the zero set of its biquaternion norm is the null cone, and the algebra inverse, composed with parity, is the conformal inversion, from which dilations and special conformal transformations are built by composition. The algebra does not carry the conformal group: it is eight real dimensions and its unit-norm group six, while the conformal group is fifteen; six of the fifteen generators are elements of $\mathbb{B}$, one more (the dilation) is realized by a non-unit element, and the remaining eight — translations and special conformal transformations — are not elements of $\mathbb{B}$ at all, none of them linear on $\mathbb{M}_-$. The standard algebraic homes of the conformal group are larger: $2\times2$ matrices over the Clifford algebra $\mathrm{Cl}_{1,3}$, or the Clifford algebra $\mathrm{Cl}_{2,4}$ of the six-dimensional conformal embedding space, whose even part has real dimension $32$ against the eight of $\mathbb{B}$. The article states these boundaries and does not smooth them over.

The conventions are those of the read-list companions. The biquaternion algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, the quaternion basis is $e_0=1,e_1,e_2,e_3$ with $e_k^2=-e_0$ and $e_1e_2=e_3$, and the scalar imaginary is $i$, commuting with the quaternion units. The subspaces are $$ \mathbb{M}_-=\{\tilde{Q}:\tilde{Q}^{*}=-\tilde{Q}\},\qquad \mathbb{M}_+=\{\tilde{Q}:\tilde{Q}^{*}=\tilde{Q}\}, $$ the anti-Hermitian (material) and Hermitian (informational) sectors, with $\mathbb{H}_{\mathbb{B}}$ the real-quaternion subspace and $\mathbb{C}_{\mathbb{B}}$ the scalar subspace. The biquaternion norm is $N(\tilde{Q})=\tilde{Q}\tilde{Q}^{\natural}=\sum_{\mu}Q_\mu^2$, and the trace pairing on the Hermitian sector is $\mathrm{Tr}(\tilde{P}\tilde{H})=2\,\mathrm{Sc}(\tilde{P}\tilde{H})$. Throughout, $c=1/\sqrt{\epsilon\mu}$ is the speed of light in the medium and $c_0$ its vacuum value. The metric on $\mathbb{M}_-$ is used in the $(ict,x,y,z)$ convention, so that $N(ict\,e_0+\mathbf{x})=-c^2t^2+|\mathbf{x}|^2$, with $\eta=\mathrm{diag}(-1,1,1,1)$ below.

The Conformal Group and Its Generators

A conformal transformation of a pseudo-Riemannian manifold is a diffeomorphism whose pullback of the metric is a positive multiple of the metric, $$ \phi^*g=\Omega^2\,g,\qquad \Omega>0 . $$ It need not preserve lengths or the interval; it preserves the null directions. In Minkowski space the globally defined conformal transformations form the group $O(2,4)$ of dimension $15$; the connected component is $SO^+(2,4)$, and its double cover is $\mathrm{Spin}(2,4)\cong SU(2,2)$, so $$ SO^+(2,4)\;\cong\;SU(2,2)/\{\pm I_4\}. $$ The conformal group contains the Poincaré group as the subgroup generated by Lorentz transformations and translations, and it is completed by the dilations and the special conformal transformations.

The generators fall into four families, fifteen in all.

Family	Symbol	Count	Finite action on $x^\mu$
Lorentz	$M_{\mu\nu}$	6	$x^\mu\mapsto\Lambda^\mu{}_\nu x^\nu$
Translations	$P_\mu$	4	$x^\mu\mapsto x^\mu+a^\mu$
Dilation	$D$	1	$x^\mu\mapsto\lambda x^\mu$
Special conformal	$\mathcal{K}_\mu$	4	$x^\mu\mapsto\dfrac{x^\mu+a^\mu x^2}{1+2a\cdot x+a^2x^2}$

The symbol $\mathcal{K}_\mu$ is used for the special conformal generator because the literature's usual $K_\mu$ is taken in this series by the boost generator $K_k=ie_k$ of the Lorentz-group companion, which carries a spatial index and is an element of the algebra. The distinction is not merely typographical below.

A concrete realization of the fifteen generators is by vector fields on the coordinates, $$ P_\mu=\partial_\mu,\qquad D=x^\nu\partial_\nu,\qquad M_{\mu\nu}=x_\mu\partial_\nu-x_\nu\partial_\mu,\qquad \mathcal{K}_\mu=2x_\mu x^\nu\partial_\nu-x^2\partial_\mu, $$ with $x^2=\eta_{\rho\sigma}x^\rho x^\sigma$. In this real form the structure constants are real, matching the convention of the Lorentz-group companion, whose brackets are $[J_j,J_k]=2\varepsilon_{jkl}J_l$, $[J_j,K_k]=2\varepsilon_{jkl}K_l$, $[K_j,K_k]=-2\varepsilon_{jkl}J_l$ for $J_k=e_k$, $K_k=ie_k$. The brackets are $$ [D,P_\mu]=-P_\mu,\qquad [D,\mathcal{K}_\mu]=\mathcal{K}_\mu, \qquad [D,M_{\mu\nu}]=0, $$ $$ [P_\mu,P_\nu]=0,\qquad [\mathcal{K}_\mu,\mathcal{K}_\nu]=0, \qquad [P_\mu,\mathcal{K}_\nu]=2(\eta_{\mu\nu}D-M_{\mu\nu}), $$ $$ [M_{\mu\nu},P_\rho]=\eta_{\nu\rho}P_\mu-\eta_{\mu\rho}P_\nu, \qquad [M_{\mu\nu},\mathcal{K}_\rho]=\eta_{\nu\rho}\mathcal{K}_\mu-\eta_{\mu\rho}\mathcal{K}_\nu, $$ together with the standard Lorentz brackets for $[M_{\mu\nu},M_{\rho\sigma}]$. These relations were recomputed on two independent polynomial test functions by direct commutator of the vector fields, with no failures.

Where the generators live. The six Lorentz generators $J_k=e_k$ and $K_k=ie_k$ are elements of $\mathbb{B}$, and the companion article builds the whole finite-dimensional Lorentz representation theory from them. The other nine generators above are differential operators on the coordinates, not elements of $\mathbb{B}$. This split — six inside the algebra, nine outside — is the theme of the sections that follow; the dilation is a partial exception, expressible by a non-unit biquaternion, and the translations and special conformal transformations are the genuinely external part.

How the Group Entered Physics

The fifteen-parameter group did not enter physics as the conformal group. In 1909–10 Harry Bateman and Ebenezer Cunningham, following Sophus Lie's work of 1871 on the relation between sphere transformations with an imaginary radius coordinate and four-dimensional conformal transformations, showed that setting the fourth coordinate to $ict$ turns Lie's sphere transformations into conformal transformations of spacetime, and that the quadratic form and Maxwell's equations are covariant under them, irrespective of a scale $\lambda$; Bateman called them spherical wave transformations. The covariance is not universal, and that is the physical content of the split above: the conformal group is the symmetry of the electrodynamic laws, whereas the Lorentz and Poincaré groups are the symmetries of all the laws of nature in inertial frames. In the parametrisation of those authors the difference is a scale, and fixing $\lambda = 1$ leaves the ten-parameter Poincaré group as the subgroup of the fifteen-parameter conformal group. (The historical literature sometimes writes "Lorentz group" where the ten-parameter subgroup is meant; the parameter count makes it the Poincaré group.) The conformal group of the plane is, in the same circle of ideas, isomorphic to the Lorentz group.

The Null Cone: Conformal Structure the Algebra Already Has

The conformal group is the group that preserves null directions, and in this framework "null" is already algebraic. A displacement $\tilde{Q}\in\mathbb{M}_-$ is null exactly when $$ N(\tilde{Q})=\tilde{Q}\tilde{Q}^{\natural}=0, $$ and the nonzero null elements are exactly the zero divisors of $\mathbb{B}$. The light cone of Minkowski space is therefore the zero-divisor cone of the algebra, whose geometry is the subject of the companion article on the null quadric and projective geometry; here only its invariance is needed. The algebra contains the object that the conformal group preserves, even though, as the later sections show, it does not contain the group.

The four families of generators act on the zero set of $N$ as follows; each statement is an elementary consequence of the multiplicativity of the biquaternion norm, and each was checked on explicit elements.

Lorentz. Rotor conjugation preserves $N$ pointwise, $$ N(\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*})=N(\tilde{\Lambda})\,N(\tilde{Q})\,N(\tilde{\Lambda}^{*})=N(\tilde{Q}), $$ since $N(\tilde{\Lambda})=e_0$ and the biquaternion norm is central. These are the isometries: the conformal transformations that leave $N$ unchanged rather than scaling it. The dilations and the special conformal transformations do not.
Dilation. The dilation $\tilde{Q}\mapsto\lambda\tilde{Q}$ scales the form, $N\mapsto\lambda^2N$. It preserves the zero set and changes the scale, which is exactly the conformal condition with constant factor.
Translation. A shift does not preserve $N$ pointwise, but it preserves the interval between two points, $$ N\bigl((\tilde{Q}+\tilde{A})-(\tilde{Y}+\tilde{A})\bigr)=N(\tilde{Q}-\tilde{Y}), $$ so it maps null displacements to null displacements. This is the parent article's observation that the interval survives translation.
Inversion. The inversion defined in the next section but one satisfies $N(I(\tilde{Q}))=1/N(\tilde{Q})$. It preserves the zero set, where it is defined, and inverts the scale.

So all four families preserve the null cone, and the invariant object of the conformal group is native to the algebra: it is the zero set of $N$. Two qualifications belong here rather than at the end. The inversion is undefined where $N(\tilde{Q})=0$, that is, on the light cone of the origin; the conformal group is only globally defined on the conformal compactification, and on $\mathbb{M}_-$ its transformations are partly defined rational maps. This article works with those rational maps and does not construct the compactification. And the invariance of the cone is a statement about the object: it does not by itself produce a group action inside $\mathbb{B}$, which is the substance of the strain.

Dilations and the Non-Unit Biquaternion

The dilation $$ \tilde{Q}\;\longmapsto\;\lambda\,\tilde{Q},\qquad \lambda>0, $$ is scalar multiplication on the material sector. It can be written as a signed inner conjugation, $$ \tilde{Q}\;\longmapsto\;\tilde{G}\,\tilde{Q}\,\tilde{G}^{*}, \qquad \tilde{G}=\sqrt{\lambda}\,e_0 , $$ and this displays both what the algebra provides and what it withholds. Three points, each recomputed on explicit elements.

The generator is not a unit-norm biquaternion. The element $\tilde{G}=\sqrt{\lambda}e_0$ has biquaternion norm $N(\tilde{G})=\lambda\,e_0\ne e_0$ for $\lambda\ne1$, so it lies outside $SL(2,\mathbb{C})$. The rotor group acts by isometries and preserves $N$; the dilation scales $N$, so it cannot be a member of that group. The dilation is realizable by the algebra but is not of the group of the algebra.

The dilation is not an algebra automorphism. Multiplication by a scalar is linear but not multiplicative: $\lambda(\tilde{Q}\tilde{Y})$ is not $(\lambda\tilde{Q})(\lambda\tilde{Y})$ for $\lambda\ne1$, a defect that was checked by direct computation on a non-commuting pair. So the dilation, unlike the rotations, cannot be read as a relabelling of the algebra's own product.

The dilation generator is not an inner derivation. Its finite form, extended to all of $\mathbb{B}$ as $\lambda$ times the identity, has trace $8\lambda\ne0$; every inner derivation $\mathrm{ad}_{\tilde{A}}=[\tilde{A},\cdot\,]$ of the eight-dimensional algebra is traceless, and the infinitesimal generator likewise has trace $8$. The trace of $\mathrm{ad}_{\tilde{A}}$ was computed in a real basis for several random $\tilde{A}$ and vanishes, against $8$ for the identity. So no element of $\mathbb{B}$ generates the dilation by commutator. This is the precise sense in which one of the fifteen parameters sits half inside the algebra: expressible, but not a group element and not an inner derivation.

Inversion and the Special Conformal Transformations

The special conformal transformations are the part of the conformal group that is nonlinear on spacetime, and they are generated by the inversion.

Inversion. For $\tilde{Q}\in\mathbb{M}_-$ with $N(\tilde{Q})\ne0$, define $$ I(\tilde{Q})=\frac{\tilde{Q}}{N(\tilde{Q})}. $$ This is the four-vector form of $x^\mu\mapsto x^\mu/x^2$, and it is algebraic: it is the quaternion conjugate followed by the algebra inverse, $$ I(\tilde{Q})=(\tilde{Q}^{\natural})^{-1}, $$ because $N(\tilde{Q}^{\natural})=N(\tilde{Q})$ (the biquaternion norm is central, so $\tilde{Q}^{\natural}\tilde{Q}=\tilde{Q}\tilde{Q}^{\natural}$) and therefore $(\tilde{Q}^{\natural})^{-1}=\overline{\tilde{Q}^{\natural}}/N(\tilde{Q}^{\natural})=\tilde{Q}/N(\tilde{Q})$. Two properties follow and were verified: $I^2=\mathrm{id}$ where defined, and $$ N\bigl(I(\tilde{Q})\bigr)=\frac{1}{N(\tilde{Q})}, $$ so $I$ maps the null cone to itself and is a conformal map with the reciprocal scale factor. Unlike the Lorentz, dilation and translation parts, the inversion is not built from the algebra's multiplication and addition alone: it uses the inverse, that is, division by the biquaternion norm.

There is a small but real distinction to record. The algebra's own inverse is $$ \tilde{Q}^{-1}=\frac{\tilde{Q}^{\natural}}{N(\tilde{Q})}, $$ which differs from $I$ by the spatial reflection $\mathbf{x}\mapsto-\mathbf{x}$, that is, by quaternion conjugation on $\mathbb{M}_-$: $$ \tilde{Q}^{-1}=P\bigl(I(\tilde{Q})\bigr),\qquad P:\,(ict,\mathbf{x})\longmapsto(ict,-\mathbf{x}). $$ The map $P$ is an improper Lorentz transformation, so it is already in the conformal group; the group generated by either choice is the same. But the two maps are different, and it is $I=\bar{(\,\cdot\,)}^{-1}$, quaternion conjugation followed by inversion, that is the standard inversion $x^\mu\mapsto x^\mu/x^2$, not the naive algebra inverse. Writing "inversion is the biquaternion inverse" without this qualification is false; the correction is exactly parity.

Special conformal transformations. A translation sandwiched between two inversions is a special conformal transformation: $$ \tilde{Q}'=I\bigl(I(\tilde{Q})+\tilde{A}\bigr) =\frac{\tilde{Q}+\tilde{A}\,N(\tilde{Q})}{1+2B(\tilde{Q},\tilde{A})+N(\tilde{A})N(\tilde{Q})}, \qquad \tilde{A}\in\mathbb{M}_-, $$ where $$ B(\tilde{P},\tilde{Q})=\tfrac12\bigl(N(\tilde{P}+\tilde{Q})-N(\tilde{P})-N(\tilde{Q})\bigr)=\sum_\mu P_\mu Q_\mu $$ is the polar form of $N$ — the complex bilinear dot product, which on $\mathbb{M}_-$ is the Minkowski inner product. In components, with $x^2=N(\tilde{Q})$ and $a\cdot x=B(\tilde{Q},\tilde{A})$, this is the standard $$ x'^\mu=\frac{x^\mu+a^\mu x^2}{1+2a\cdot x+a^2x^2}. $$ The identity of the two expressions was checked directly on random four-vectors $\tilde{Q},\tilde{A}$, and the two displayed formulas agree. This answers, in part, the open question of the Poincaré article: the translation does appear as a product of generalized inversions, but the product is the special conformal transformation, not the translation, and it is not linear in $\tilde{Q}$. The translation itself remains an additive shift.

Two qualifications belong to the formula. First, it is a fractional (nonlinear) expression in $\tilde{Q}$, not the two-sided linear map $\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$ that carries the Lorentz group; the algebraic operation that produces it is inversion, and inversion is a rational map, not a linear one. Second, it is only partly defined: the inversion is singular on the null cone of the origin, and the denominator $1+2a\cdot x+a^2x^2$ vanishes on a cone as well. Globally, these are transformations of the conformal compactification; on $\mathbb{M}_-$ they are defined off a null set.

The generator. Expanding the special conformal formula to first order in $a$, $$ \delta x^\mu=a^\mu x^2-2x^\mu(a\cdot x)=-a^\nu\,\mathcal{K}_\nu x^\mu, $$ with $\mathcal{K}_\nu=2x_\nu x^\rho\partial_\rho-x^2\partial_\nu$. This is the vector field of the generator table; the algebra produces the finite transformation but not an element of $\mathbb{B}$ that generates it.

Where the Algebra Strains

The obstructions that keep the conformal group out of the algebra can now be collected. None is a matter of insufficient cleverness in choosing a parametrization; each is structural.

Dimension. The biquaternion algebra has real dimension $8$, and its unit-norm subgroup $SL(2,\mathbb{C})$ — the group that acts as the Lorentz group — has real dimension $6$. The conformal group has dimension $15$. No faithful assignment of unit-norm biquaternions to conformal elements exists, for the same counting reason that blocks the translations in the Poincaré article. In Lie-algebra terms, $\mathrm{SO}(2,4)$ is fifteen-dimensional and cannot embed as a subalgebra of $\mathbb{B}$ under the commutator bracket, which is eight-dimensional. No arrangement of the existing algebra closes the gap.

Linearity. The natural self-action of the algebra is the linear map $\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$; it preserves $N$ pointwise, hence produces isometries only, never dilations, and its group is six-dimensional. For a rotation this map is conjugation by a unit real quaternion and is an inner automorphism; for a boost it is the twisted map $\tilde{B}\tilde{Q}\tilde{B}$ (since $\tilde{B}^{*}=\tilde{B}$), which is linear but not multiplicative and so is not an algebra automorphism at all. Either way it is linear in $\tilde{Q}$, and the special conformal transformations are not. Classifying the fifteen generators by how they can be realized:

the six Lorentz generators are elements of $\mathbb{B}$;
the dilation is a non-unit element and a non-inner, nonzero-trace linear map;
the four translations are affine, hence not linear on $\mathbb{M}_-$;
the four special conformal generators are fractional, hence not linear at all.

The size of the linearly realized part. The conformal transformations that act linearly or affinely on $\mathbb{M}_-$ are exactly the similarities $x^\mu\mapsto\lambda\Lambda^\mu{}_\nu x^\nu+a^\mu$, whose group — Lorentz (6) plus dilation (1) plus translations (4) — has dimension $11$. The remaining four generators, the special conformal ones, are precisely the nonlinear part. The algebra supplies all eleven: the Lorentz factor by multiplication of rotors, the dilation by a non-unit scaling, the translations by addition. The last four it reaches only through the nonlinear inversion. That is the sharp form of the boundary.

The second sector. The Lorentz action extends to the whole algebra: $\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$ is a linear map of $\mathbb{B}$ that restricts to rotor conjugation on $\mathbb{M}_-$ and preserves $\mathbb{M}_+$ as well (the conjugate of a Hermitian element is Hermitian). Conformal operations behave less uniformly. The inversion does preserve each sector separately: $N$ is real on $\mathbb{M}_+$ and on $\mathbb{M}_-$, so the scalar factor $\tilde{Q}/N(\tilde{Q})$ keeps the Hermitian or anti-Hermitian character. But the translations are $\mathbb{M}_-$-valued and the special conformal transformations are anchored there, so the full conformal group has no canonical action on the informational sector $\mathbb{M}_+$. Whether $\mathbb{M}_+$ is meant to be conformally invariant, and with what weight, is not determined by the algebra. This is a gap, not a result, and it is left visible.

Two Larger Homes in the Literature

The conformal group is a standard object, and its standard algebraic homes are larger than $\mathbb{B}$. Two are worth naming, because they locate exactly what the biquaternion algebra would have to acquire.

Vahlen matrices. The Ahlfors–Vahlen construction realizes the conformal group of $\mathbb{R}^{p,q}$ by $2\times2$ matrices $$ \begin{pmatrix} a & b\\ c & d\end{pmatrix} $$ whose entries lie in the Clifford algebra $\mathrm{Cl}_{p,q}$, are products of vectors, and satisfy the conformality conditions that make the fractional action $$ \tilde{Q}\;\longmapsto\;(aX+b)(cX+d)^{-1} $$ reduce to a conformal map of vectors. For Minkowski space the entries live in $\mathrm{Cl}_{1,3}$, of real dimension $16$ — twice the eight of $\mathbb{B}$. The biquaternion algebra is the even subalgebra, $\mathbb{B}=\mathrm{Cl}^+_{1,3}$, and the natural question — whether the Vahlen conditions can be imposed with entries in the even subalgebra alone, giving a reduced Vahlen group — is a technical point that this article does not settle. It is stated here as the precise form of "the algebra is too small."

The six-dimensional embedding. Alternatively, the conformal group is a rotation group of a six-dimensional space. The conformal compactification of $\mathbb{R}^{1,3}$ is the projectivized null cone of $\mathbb{R}^{2,4}$, exactly as Minkowski space is a real slice of the complexified rotation picture of the Lorentz-transformation companion; the conformal group is $O(2,4)$, realized by rotations of that six-dimensional space. In Clifford terms this is $\mathrm{Cl}_{2,4}$: its bivectors span the fifteen-dimensional conformal algebra, and its even subalgebra, the home of the conformal rotors, has real dimension $32$. The biquaternion algebra, as the even subalgebra of the four-dimensional Clifford algebra $\mathrm{Cl}_{1,3}$, is eight-dimensional. The conformal rotor algebra is four times larger. This is the quantitative content of the Poincaré article's remark that any enlargement forced by the translations must be strictly larger than $\mathbb{B}$.

The Clifford-group reading, and dilations without a Weyl field. The six-dimensional embedding is sometimes stated as its own result: the conformal group of four-dimensional spacetime emerges as a subgroup of the Clifford group [Castro and Pavšič, "Clifford Algebra of Spacetime and the Conformal Group," Int. J. Theor. Phys. 42 (2003) 1693, arXiv:hep-th/0203194]. The standard part is the realization of the fifteen generators in the Clifford algebra of spacetime: with $e_a = (\gamma_\mu, \gamma_5, 1)$ the six-tuplet, $L_{ab} = -\tfrac{i}{4}[e_a,e_b]$ gives the Lorentz generators $M_{\mu\nu}$, the dilation $D = L_{65}$, the translations $P_\mu = L_{5\mu}+L_{6\mu}$, and the special conformal generators, so the whole conformal algebra is spanned by commutators of $1$, $\gamma_\mu$ and $\gamma_5$ — the same $\mathrm{Cl}_{2,4}$ statement as above, and the authors' own comment is that this "might look mathematically trivial". What the reference adds is where that six-dimensional space sits: it is the subspace of the Clifford space spanned by the scalar unit, the four vector generators and the pseudoscalar unit, with metric signature $(+,-,-,-,-,+)$ and coordinates $(\eta^\mu,\eta^5,\eta^6)$ replacing $(x^\mu,\tilde\sigma,\sigma)$, so that the conformal group acts linearly there and on the cone $\eta^a\eta_a = 0$ reproduces the conformal transformations of Minkowski space. The remaining polyvector coordinates — the bivectors $x^{\mu\nu}$ and the pseudovectors $\tilde{x}^\mu$ — lie outside that subspace and take no part. The same passage adds, and this is speculative, a physical reading of the dilations: the authors argue that extended objects in this arena change size "in the absence of forces and the Weyl gauge field of dilations", moving dilationally because of inertia rather than because a compensating gauge field has been introduced, citing Pavšič's own earlier work. That is a proposal about why scale changes occur, not a derivation, and it is recorded here as the programme's claim, not as a result of the algebra.

Twistors. A third realization uses the same vector space as the algebra but a different structure. The double cover $SU(2,2)$ of the conformal group acts linearly on twistor space $T=S\oplus\bar{S}\cong\mathbb{C}^4$. As complex vector spaces, $T$ and $\mathbb{B}$ are both $\mathbb{C}^4$; but the form that $SU(2,2)$ preserves is a Hermitian form of signature $(2,2)$, whereas the biquaternion norm $N$ is a complex symmetric bilinear form. The spaces agree; the forms do not. This is the precise sense in which the conformal group, when it acts on the same four-dimensional vector space, enters as the unitary group of a form and not as the unit group of a norm — a different kind of group from the one the framework uses. The companion article on twistors develops this in full, with the incidence relation and the Klein correspondence; the point here is only that the twistor realization does not place the conformal group inside $\mathbb{B}$, it places it on $\mathbb{B}$ read as a new module with a new form.

Summary

The conformal group of Minkowski space is the fifteen-parameter group $SO(2,4)$, double-covered by $SU(2,2)$, generated by the six Lorentz generators $M_{\mu\nu}$, the four translations $P_\mu$, the single dilation $D$, and the four special conformal generators $\mathcal{K}_\mu$. Its brackets were recomputed here on explicit test functions: $[D,P_\mu]=-P_\mu$, $[D,\mathcal{K}_\mu]=\mathcal{K}_\mu$, $[P_\mu,\mathcal{K}_\nu]=2(\eta_{\mu\nu}D-M_{\mu\nu})$, with the translations and special conformal generators abelian and the Lorentz part the companion article's.

The biquaternion algebra carries the conformal structure but not the conformal group. What it carries: the null cone, which is the zero-divisor cone $\{N=0\}$, is the invariant object of the conformal group, and every generator preserves it; the inversion is algebraic, $I(\tilde{Q})=(\tilde{Q}^{\natural})^{-1}=\tilde{Q}/N(\tilde{Q})$, with $I^2=\mathrm{id}$ and $N(I(\tilde{Q}))=1/N(\tilde{Q})$; the dilation is the signed inner conjugation by the non-unit element $\sqrt{\lambda}e_0$; and the special conformal transformation is the composition $I\circ T_{\tilde{A}}\circ I$, with the explicit form $$ \tilde{Q}'=\frac{\tilde{Q}+\tilde{A}\,N(\tilde{Q})}{1+2B(\tilde{Q},\tilde{A})+N(\tilde{A})N(\tilde{Q})}, $$ which reproduces $x'^\mu=(x^\mu+a^\mu x^2)/(1+2a\cdot x+a^2x^2)$.

What it does not carry: the group. The algebra is eight real dimensions and its unit group six, against the conformal group's fifteen; the six Lorentz generators are elements of $\mathbb{B}$, the dilation is a non-unit element and a non-inner, nonzero-trace linear map, the four translations are affine and the four special conformal transformations are fractional. The transformations realized linearly or affinely on $\mathbb{M}_-$ form the eleven-dimensional similitude group, and the four remaining generators are exactly the nonlinear part. The standard homes of the full group are larger: Vahlen matrices over $\mathrm{Cl}_{1,3}$ (real dimension $16$), conformal geometric algebra in $\mathrm{Cl}_{2,4}$ (even part real dimension $32$), or $SU(2,2)$ acting on a $\mathbb{C}^4$ equipped with a signature-$(2,2)$ Hermitian form that the biquaternion norm is not. The conformal group is expressed by the algebra — by its biquaternion norm's zero set, its inverse, and its four-vector module — and is not contained in it.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	Biquaternion algebra, real dimension $8$
$e_0=1,e_1,e_2,e_3$	Quaternion basis, $e_k^2=-e_0$, $e_1e_2=e_3$
$i$	Scalar imaginary, $i^2=-1$, commuting with $e_k$
$\mathbb{M}_-$, $\mathbb{M}_+$	Anti-Hermitian (material) and Hermitian (informational) sectors
$\mathbb{H}_{\mathbb{B}}$, $\mathbb{C}_{\mathbb{B}}$	Real-quaternion and scalar subspaces
$N(\tilde{Q})=\tilde{Q}\tilde{Q}^{\natural}=\sum_\mu Q_\mu^2$	Biquaternion norm; its zero set is the null cone
$B(\tilde{P},\tilde{Q})=\sum_\mu P_\mu Q_\mu$	Polar form of $N$; Minkowski inner product on $\mathbb{M}_-$
$\tilde{\Lambda}$, $J_k=e_k$, $K_k=ie_k$	Lorentz rotor and its rotation/boost generators (in $\mathbb{B}$)
$\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$	Rotor conjugation (four-vector action)
$SL(2,\mathbb{C})\cong\{N=e_0\}$	Unit-norm biquaternions, Lorentz double cover, real dimension $6$
$M_{\mu\nu}$, $P_\mu$, $D$, $\mathcal{K}_\mu$	Conformal generators: Lorentz (6), translations (4), dilation (1), special conformal (4)
$I(\tilde{Q})=(\tilde{Q}^{\natural})^{-1}=\tilde{Q}/N(\tilde{Q})$	Inversion; $I^2=\mathrm{id}$, $N(I(\tilde{Q}))=1/N(\tilde{Q})$
$\tilde{Q}^{-1}=\tilde{Q}^{\natural}/N(\tilde{Q})=P(I(\tilde{Q}))$	Algebra inverse = inversion followed by parity
$I\circ T_{\tilde{A}}\circ I$	Special conformal transformation
$SO^+(2,4)$, $SU(2,2)$	Conformal group (dim $15$) and its double cover
$\mathrm{Cl}_{1,3}=\mathrm{Cl}^+_{1,3}\oplus\mathrm{Cl}^-_{1,3}$, $\mathbb{B}=\mathrm{Cl}^+_{1,3}$	Clifford algebra of Minkowski space, real dimension $16$
$\mathrm{Cl}_{2,4}$	Conformal embedding Clifford algebra; bivectors $\cong\mathrm{SO}(2,4)$, even part real dimension $32$
$\mathrm{Tr}(\tilde{P}\tilde{H})=2\,\mathrm{Sc}(\tilde{P}\tilde{H})$	Trace pairing on the Hermitian sector
$c=1/\sqrt{\epsilon\mu}$, $c_0$	Speed of light in the medium; in vacuum