The Lorentz Group as Biquaternion Norm Automorphisms

Introduction

The group that acts on the material sector of the biquaternion framework has so far been introduced in two ways: as the group of unit-norm biquaternions, and as the group of rotor conjugations they generate. Both descriptions take the group as given and derive its action. This article takes the opposite route. It asks what group the algebra forces when the biquaternion norm is regarded as the structure to be preserved, and shows that the answer is the Lorentz group, with the rotor description recovered as the coordinate form of the automorphisms.

The point of the automorphism reading is that the biquaternion norm is not an extra structure laid on the algebra. It is the algebra's own multiplicative quadratic form,

$$ N(\tilde{Q}) = \tilde{Q}\overline{\tilde{Q}} = \sum_{\mu=0}^{3}Q_\mu^2, $$

and it satisfies

$$ N(\tilde{Q}\tilde{R}) = N(\tilde{Q})\,N(\tilde{R}) $$

for all biquaternions $\tilde{Q},\tilde{R}$. A multiplicative quadratic form on a four-dimensional algebra is a rare object, and the group that preserves it is thereby tied to the algebra's multiplication rather than imposed from outside. The automorphism group of the form on the complex algebra is $O(4,\mathbb{C})$; the automorphisms that also preserve the algebra's real structure — the anti-Hermitian material slice — form the Lorentz group.

Four statements organize the article. The first three are the levels at which the identification by the form can be read; the fourth is the level supplied by the product.

• At the complex level, $N$ is a nondegenerate symmetric form on $\mathbb{B}\cong\mathbb{C}^4$, and the automorphism group is the complex orthogonal group $O(4,\mathbb{C})$. Via the determinant realization $N=\det$, the connected component is $SO(4,\mathbb{C})\cong(SL(2,\mathbb{C})\times SL(2,\mathbb{C}))/\{\pm(e_0,e_0)\}$, acting by $\tilde{Q}\mapsto\tilde{A}\tilde{Q}\tilde{B}^{-1}$.
• At the real level, the restricted form on the anti-Hermitian slice is the Minkowski form $\eta = \mathrm{diag}(-1,+1,+1,+1)$, and the automorphisms preserving the slice are the real orthogonal maps of signature $(3,1)$, with identity component the restricted Lorentz group $SO^+(1,3)$.
• At the rotor level, the identity component is exactly the group of conjugations $\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$ by unit-norm biquaternions, and the correspondence is two-to-one.
• At the algebra level, the automorphisms that preserve the product and not only the form are the inner ones, and they are the diagonal $SO(3,\mathbb{C})\cong PGL(2,\mathbb{C})$ inside $SO(4,\mathbb{C})$. As a real group this is again the restricted Lorentz group, and it acts as the Lorentz group on a real coordinate system that is bilinear in the coordinates of a complex 3-space, with the causal structure following from an identity rather than from a postulate. That route is the source's, and the section The Algebra Automorphisms: the Diagonal, and an Induced Causality treats it as the middle case between the norm and the slice.

Boundaries. This is a group-theoretic and geometric article. The spinor module, its one-sided action, and the representation theory of the group belong to the sibling category on relativistic quantum theory and to the companion article The Spinor Module in Biquaternionic Form and Its Lorentz Action; they are not developed here. The topology of the cover, and the composition law of boosts in detail, belong to the companion article The Two-Sheeted Cover and the Topology of Boosts in Biquaternionic Form. The structure and the finite-dimensional representations of the group as such are treated in The Lorentz Group in Biquaternionic Form — Structure and Representations; this article's subject is the characterization of the group by the form.

Conventions. We use those of the read list unchanged. The algebra is $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, with quaternion basis $e_0 = 1, e_1, e_2, e_3$, $e_k^2 = -e_0$ and $e_je_k = -\delta_{jk}e_0 + \varepsilon_{jkl}e_l$, and central scalar imaginary $i$. The subspaces are $\mathbb{M}_-$ (anti-Hermitian: imaginary scalar, real vector — the material sector), $\mathbb{M}_+$ (Hermitian: real scalar, imaginary vector — the informational sector), $\mathbb{H}_{\mathbb{B}}$ (real quaternions) and $\mathbb{C}_{\mathbb{B}} = \mathrm{span}_\mathbb{R}\{e_0, ie_0\}$ (the center). The conjugations are ${}^{\natural}$ (quaternion), $\bar{\cdot}$ (complex), ${}^{*} = ({}^{\natural})^{\,*}$ (Hermitian) and ${}^\flat = -{}^{*}$ (anti-Hermitian). The biquaternion norm is $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural} = \sum_\mu Q_\mu^2$ — level 1, the identity $\mathrm{diag}(+1,+1,+1,+1)$ on $\mathbb{C}$ — and its restriction to the real material slice is the level-2 form $\eta = \mathrm{diag}(-1,+1,+1,+1)$. The matrix realization is $\Phi:\mathbb{B}\to M_2(\mathbb{C})$ with $\Phi(e_0) = I_2$, $\Phi(e_k) = -i\sigma_k$, $\Phi(i) = iI_2$. The material coordinate is $\tilde{Q} = ict\,e_0 + \mathbf{x}$. Throughout, $c = 1/\sqrt{\epsilon\mu}$ is the speed of light in the medium and $c_0$ its vacuum value. The trace formula is $\mathrm{Tr}(\tilde{P}\tilde{H}) = 2\,\mathrm{Sc}(\tilde{P}\tilde{H})$.

The Biquaternion Norm as a Quadratic Form

The biquaternion norm is the quadratic map

$$ N:\ \mathbb{B}\longrightarrow\mathbb{C}, \qquad N(\tilde{Q}) = \tilde{Q}\overline{\tilde{Q}} = Q_0^2 + Q_1^2 + Q_2^2 + Q_3^2 , $$

whose polarization is the symmetric bilinear form

$$ B(\tilde{Q},\tilde{R}) = \tfrac12\left[N(\tilde{Q}+\tilde{R}) - N(\tilde{Q}) - N(\tilde{R})\right] = Q_0R_0 + Q_1R_1 + Q_2R_2 + Q_3R_3 . $$

In the coordinate basis $(e_0,e_1,e_2,e_3)$ its matrix is the identity,

$$ B(\tilde{Q},\tilde{R}) = \sum_\mu Q_\mu R_\mu , \qquad G = \mathrm{diag}(+1,+1,+1,+1) , $$

so $N$ is a nondegenerate complex quadratic form of maximal Witt index on $\mathbb{C}^4$. It is not positive definite over $\mathbb{C}$ — no complex form is — and it is isotropic: it has nonzero null vectors, which are the zero divisors of the companion article The Light Cone as the Biquaternion Zero-Divisor Cone.

Two properties make $N$ the algebra's own form rather than a form on a vector space.

Multiplicativity. For all $\tilde{Q},\tilde{R}$,

$$ N(\tilde{Q}\tilde{R}) = \tilde{Q}\tilde{R}\,\overline{\tilde{Q}\tilde{R}} = \tilde{Q}\tilde{R}\,\overline{\tilde{R}}\,\overline{\tilde{Q}} = \tilde{Q}\,N(\tilde{R})\,\overline{\tilde{Q}} = N(\tilde{R})\,\tilde{Q}\overline{\tilde{Q}} = N(\tilde{Q})\,N(\tilde{R}), $$

where the fourth equality uses that $N(\tilde{R})$ is a complex scalar and therefore central in $\mathbb{B}$. The biquaternion norm is a homomorphism of multiplicative monoids from $(\mathbb{B},\cdot)$ to $(\mathbb{C},\cdot)$.

Compatibility with the real structure. The conjugations act on the form by

$$ N(\tilde{Q}^{*}) = \overline{N(\tilde{Q})} = N(\tilde{Q})^{*}, \qquad N(\tilde{Q}^{\natural}) = N(\tilde{Q}), $$

so the form is real on the Hermitian and anti-Hermitian slices. This is what will allow the complex form to restrict to a real form of Minkowski signature.

Automorphisms of the form. An automorphism of $(\mathbb{B},N)$ is an invertible $\mathbb{C}$-linear map $T$ with

$$ N(T\tilde{Q}) = N(\tilde{Q}) \qquad\text{for all }\tilde{Q}\in\mathbb{B}. $$

Since $N$ is nondegenerate with matrix the identity, the group of such maps is the complex orthogonal group

$$ O(4,\mathbb{C}) = \{\,T\in GL_4(\mathbb{C}) : T^{T}T = I_4\,\}, $$

of complex dimension six, with connected component $SO(4,\mathbb{C}) = O(4,\mathbb{C})\cap SL_4(\mathbb{C})$. This is the largest group the form alone defines. The Lorentz group will be a real form of it, selected by the requirement that the real structure of the algebra be preserved.

The Determinant Realization and the Complex Group

The matrix realization makes the form and its automorphisms concrete. With $\Phi$ as in the conventions,

$$ \Phi(\tilde{Q}) = \begin{pmatrix} Q_0 - iQ_3 & -iQ_1 - Q_2\\ -iQ_1 + Q_2 & Q_0 + iQ_3 \end{pmatrix}, \qquad \det\Phi(\tilde{Q}) = Q_0^2+Q_1^2+Q_2^2+Q_3^2 = N(\tilde{Q}). $$

The biquaternion norm is the determinant, and $\Phi$ is an isomorphism of $\mathbb{C}$-algebras. The verification of the determinant identity is a direct expansion; it was also checked numerically on random biquaternions, with $\det\Phi(\tilde{Q})$ and $N(\tilde{Q})$ agreeing to machine precision.

The determinant is a quadratic form on the four-dimensional space $M_2(\mathbb{C})$, and its automorphism group is classical. Consider the map

$$ T_{\tilde{A},\tilde{B}}:\ \tilde{Q}\ \longmapsto\ \tilde{A}\,\tilde{Q}\,\tilde{B}^{-1}, \qquad \tilde{A},\tilde{B}\in GL_2(\mathbb{C}). $$

It is invertible and $\mathbb{C}$-linear, and its effect on the form is

$$ N(T_{\tilde{A},\tilde{B}}\tilde{Q}) = \det\!\left(\tilde{A}\tilde{Q}\tilde{B}^{-1}\right) = \frac{\det\tilde{A}}{\det\tilde{B}}\,\det\tilde{Q} = \frac{\det\tilde{A}}{\det\tilde{B}}\,N(\tilde{Q}). $$

Preservation of $N$ therefore requires $\det\tilde{A} = \det\tilde{B}$, and one may normalize both to unit determinant; the surviving pairs are $(\tilde{A},\tilde{B})\in SL(2,\mathbb{C})\times SL(2,\mathbb{C})$. The kernel of the assignment $(\tilde{A},\tilde{B})\mapsto T_{\tilde{A},\tilde{B}}$ is the set of pairs acting trivially, $T_{\tilde{A},\tilde{B}} = \mathrm{id}$, which is

$$ \ker = \{\,(\lambda I_2,\lambda I_2) : \lambda\in\mathbb{C}^\times\,\}, \qquad \lambda^2 = 1 \ \text{ under the determinant normalization}, \qquad \ker = \{\pm(e_0,e_0)\} . $$

Hence

$$ \boxed{\; SO(4,\mathbb{C}) \;\cong\; \frac{SL(2,\mathbb{C})\times SL(2,\mathbb{C})}{\{\pm(e_0,e_0)\}} \;} \qquad \text{(complex dimension 3 + 3 = 6).} $$

The two factors are the two chiral halves of the complexified rotation group; in the algebra they correspond to left and right multiplication. This is the complex group of the biquaternion norm. It is not the Lorentz group: it acts on the complexified four-vector space, and its two $SL(2,\mathbb{C})$ factors are independent.

The Real Slice and the Minkowski Form

The Lorentz group is selected by the algebra's real structure. The anti-Hermitian slice is

$$ \mathbb{M}_- = \{\tilde{Q}\in\mathbb{B} : \tilde{Q}^{*} = -\tilde{Q}\}, $$

a four-real-dimensional subspace, with the real basis $\{ie_0, e_1, e_2, e_3\}$. Writing

$$ \tilde{Q} = i x_0 e_0 + x_1e_1 + x_2e_2 + x_3e_3, \qquad x_\mu\in\mathbb{R}, $$

the biquaternion norm is real and indefinite:

$$ N(\tilde{Q}) = (ix_0)^2 + x_1^2 + x_2^2 + x_3^2 = -x_0^2 + \mathbf{x}^2 . $$

The restriction of $N$ to $\mathbb{M}_-$, in the real coordinates, is therefore the quadratic form with matrix

$$ \eta = \mathrm{diag}(-1,+1,+1,+1), $$

the level-2 Minkowski form of the series. Identifying $x_0 = ct$, it is the interval. The same computation on the other distinguished slices gives the full picture of the form's real restrictions:

The complex form $\mathrm{diag}(+1,+1,+1,+1)$ has real forms of both signatures; the two Hermitian-type slices carry the Minkowski real form, the real-quaternion slice the Euclidean one, and the two are exchanged by multiplication by $i$. The physical slice is $\mathbb{M}_-$.

The real structure is not decoration: it is what tells the complex group which real form of the algebra is physical. A $\mathbb{C}$-linear automorphism of the complexification need not map $\mathbb{M}_-$ to itself. The automorphisms that do are the ones the physical theory uses.

The Lorentz Group as the Automorphisms of the Slice

Consider the automorphisms of $N$ that preserve the real slice:

$$ \mathcal{G} = \{\,T\in O(4,\mathbb{C}) : T(\mathbb{M}_-)\subseteq\mathbb{M}_-\,\}. $$

On $\mathbb{M}_-$ the map $T$ is a real-linear transformation, and the condition

$$ N(T\tilde{Q}) = N(\tilde{Q})\ \text{ for all }\tilde{Q}\in\mathbb{M}_- $$

says exactly that $T$ preserves the Minkowski form $\eta = \mathrm{diag}(-1,+1,+1,+1)$. Hence

$$ \mathcal{G} \;\cong\; O(1,3), $$

the full Lorentz group including the discrete reflections and the time-reversal and parity components. The connected component of the identity is the restricted Lorentz group $SO^+(1,3)$: the transformations that are proper (determinant $+1$) and orthochronous (preserve the time direction). This is the group the algebra supplies at the real level: the automorphisms of the biquaternion norm that respect the material slice.

The discrete components are automorphisms of the form and are not rotor conjugations; they include spatial reflection and time reversal, which are outer with respect to the rotor group. The rotor group covers only the identity component, which is why a spinor or a rotor is not by itself sensitive to orientation-reversing transformations.

The Rotor Realization of the Automorphisms

The identity component has an explicit algebraic form. A Lorentz rotor is a unit-norm biquaternion,

$$ \tilde{\Lambda}\in\mathbb{B}, \qquad \tilde{\Lambda}\overline{\tilde{\Lambda}} = e_0 , $$

and it acts by conjugation,

$$ \pi(\tilde{\Lambda}):\ \tilde{Q}\ \longmapsto\ \tilde{\Lambda}\,\tilde{Q}\,\tilde{\Lambda}^{*} . $$

Three properties, each a one-line verification, are the content of the identification.

It preserves the slice. If $\tilde{Q}^{*} = -\tilde{Q}$ then

$$ \left(\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}\right)^\dagger = \tilde{\Lambda}\tilde{Q}^{*}\tilde{\Lambda}^{*} = -\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}, $$

so the image is again anti-Hermitian.

It preserves the form. By multiplicativity,

$$ N(\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}) = N(\tilde{\Lambda})\,N(\tilde{Q})\,N(\tilde{\Lambda}^{*}) = 1\cdot N(\tilde{Q})\cdot\overline{N(\tilde{\Lambda})} = N(\tilde{Q}), $$

using $N(\tilde{\Lambda}) = 1$ and $N(\tilde{\Lambda}^{*}) = \overline{N(\tilde{\Lambda})} = 1$.

It defines a homomorphism. Since $\pi(\tilde{\Lambda}_1)\pi(\tilde{\Lambda}_2) = \pi(\tilde{\Lambda}_1\tilde{\Lambda}_2)$ and $(\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*})^{*}$ reuses the same $\tilde{\Lambda}$ on both sides, the assignment is a continuous group homomorphism

$$ \pi:\ SL(2,\mathbb{C}) = \{\tilde{\Lambda} : N(\tilde{\Lambda}) = 1\} \longrightarrow SO^+(1,3). $$

Its kernel is the set of rotors acting trivially on every $\tilde{Q}\in\mathbb{M}_-$. Since $-e_0$ is central,

$$ (-e_0)\,\tilde{Q}\,(-e_0)^{*} = \tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}\big|_{\tilde{\Lambda} = -e_0} = (-e_0)\tilde{Q}(-e_0) = \tilde{Q}, $$

so $\pm e_0$ act identically, and no other element does; hence

$$ \ker\pi = \{\pm e_0\}\cong\mathbb{Z}/2\mathbb{Z}, \qquad SO^+(1,3)\cong SL(2,\mathbb{C})/\{\pm e_0\}. $$

The map $\pi$ is surjective onto $SO^+(1,3)$; that every proper orthochronous Lorentz transformation of $\mathbb{M}_-$ arises as a rotor conjugation is the standard theorem that $SL(2,\mathbb{C})$ is the double cover of the restricted Lorentz group, and it is cited here as standard rather than re-derived. The identification of the previous section is therefore complete at the level of the identity component:

$$ \boxed{\; SO^+(1,3)\;\cong\;SL(2,\mathbb{C})/\{\pm e_0\} \;=\;\{\pi(\tilde{\Lambda}): N(\tilde{\Lambda})=1\}/\ker\pi .\;} $$

Verification. The two preservation properties and the kernel statement were checked numerically. Over three hundred random unit-norm rotors $\tilde{\Lambda} = R\,B$ (a rotation times a boost, normalized) applied to random anti-Hermitian $\tilde{Q}$, the image was always anti-Hermitian and $|N(\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}) - N(\tilde{Q})|$ was zero to machine precision. For a random rotor the identity $\pi(-\tilde{\Lambda}) = \pi(\tilde{\Lambda})$ held exactly.

A word on the relation between the complex and the real descriptions is in order, because the two groups have the same complexification. The complex group $SO(4,\mathbb{C})$ has complex dimension six; the real Lorentz group $SO(1,3)$ also has real dimension six, and is a real form of it. The rotor group $SL(2,\mathbb{C})$ is a six-real-dimensional group that doubly covers the identity component. The real structure is what reduces the first to the second; the map $\pi$ is what realizes the second by conjugation with unit-norm elements.

The Algebra Automorphisms: the Diagonal, and an Induced Causality

The automorphisms used so far preserve the form. The algebra has a smaller automorphism group, the maps that also preserve the product, and the difference between the two is the subject of open question 1 below. That difference has a sharp algebraic description, and a neighbouring research programme — Kassandrov's biquaternionic algebrodynamics, in Further Reading — builds on it a derivation of Minkowski geometry and of its causal structure rather than treating it as a curiosity. The group-theoretic content of this section is standard; the construction and its consequences are the source's, and the checks are this article's.

The algebra automorphisms are the diagonal of the form automorphisms. By Skolem–Noether (companion article Biquaternion Automorphisms and Derivations) every $\mathbb{C}$-algebra automorphism of $\mathbb{B}\cong M_2(\mathbb{C})$ is inner, $\tilde{Q}\mapsto\tilde{M}\tilde{Q}\tilde{M}^{-1}$, and the group is

$$ \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \;\cong\; PGL(2,\mathbb{C}) \;\cong\; PSL(2,\mathbb{C}) \;\cong\; SO(3,\mathbb{C}), $$

of complex dimension three. In the parametrization $\tilde{Q}\mapsto\tilde{A}\tilde{Q}\tilde{B}^{-1}$ of the norm automorphisms above — where preservation of $N$ required only $\det\tilde{A}=\det\tilde{B}$ — this is the diagonal $\tilde{A}=\tilde{B}=\tilde{M}$, and the determinant condition then holds automatically. Since the full complex group is $SO(4,\mathbb{C})\cong(SL(2,\mathbb{C})\times SL(2,\mathbb{C}))/\{\pm(e_0,e_0)\}$,

$$ SO(3,\mathbb{C}) \;\cong\; \frac{SL(2,\mathbb{C})}{\{\pm e_0\}} \;\subset\; SO(4,\mathbb{C}), \qquad \text{the diagonal}. $$

Two consequences are worth separating. The algebra automorphisms are exactly the norm automorphisms with the two chiral factors locked together — the ones that preserve the multiplication and not merely the quadratic form. And it is the diagonal, not an arbitrary pair, that is real-isomorphic to the Lorentz group: $SO(3,\mathbb{C})$ and $SO^+(1,3)$ are both $PSL(2,\mathbb{C})$, connected of real dimension six, hence the same real group. The group recovered from the algebra is thus the restricted Lorentz group, whereas the group recovered from the form alone is the larger $SO(4,\mathbb{C})$; the source reads this as evidence that the algebra, not the form, is the primitive object.

What it acts on, and what it preserves. The inner automorphism fixes $\mathrm{tr}\,\tilde{Q}$ and therefore the scalar coordinate $z_0$, and it acts on the traceless part $\tilde{V}=\tilde{Q}-\tfrac12(\mathrm{tr}\tilde{Q})e_0$ by $\tilde{V}\mapsto\tilde{M}\tilde{V}\tilde{M}^{-1}$, preserving $\det\tilde{V}$. The traceless part is three-dimensional over $\mathbb{C}$, and on it the action is the complex-orthogonal one: with $x=(z_1,z_2,z_3)$,

$$ x \longmapsto Rx , \qquad R^TR = I_3 , \qquad \det R = 1 , $$

preserving the complex quadratic form $\sigma = z_1^2+z_2^2+z_3^2 = -\det\tilde{V}$. This $\sigma$ is the source's "3D complex metric". Stronger invariants come from the trace: $\det\tilde{Q}$ is preserved by the inner action, so $z_0$ and $\sigma = z_0^2-\det\tilde{Q}$ are invariants of the algebra automorphisms, which is why the whole construction is built from $z_0$ and $\sigma$.

The source's matrix realization differs from the $\Phi$ of the conventions above. It is the assignment

$$ \tilde{Q}\ \longmapsto\ \begin{pmatrix} z_0+z_3 & z_1-iz_2\\ z_1+iz_2 & z_0-z_3 \end{pmatrix}, \qquad \det = z_0^2-z_1^2-z_2^2-z_3^2 , $$

whose determinant is the source's form rather than $N=\sum_\mu Q_\mu^2$. The two forms are $\mathbb{C}$-equivalent — over $\mathbb{C}$ the substitution $z_a\mapsto iz_a$ carries one to the other, and all nondegenerate complex forms of a given dimension are equivalent — so the group, the invariant $\sigma$ and the action are the same objects as those above. Only $(\mathbb{C}^3,\sigma)$ enters what follows.

Bilinear real coordinates. Write $z_a=p_a+iq_a$ with $\mathbf{p},\mathbf{q}$ real 3-vectors, and set

$$ T = |\mathbf{p}|^2+|\mathbf{q}|^2 , \qquad \mathbf{X} = 2\,\mathbf{p}\times\mathbf{q} . $$

Then $\operatorname{Re}\sigma = |\mathbf{p}|^2-|\mathbf{q}|^2$ and $\operatorname{Im}\sigma = 2\,\mathbf{p}\cdot\mathbf{q}$, so

$$ S^2 := \sigma\sigma^{*} = \bigl(|\mathbf{p}|^2-|\mathbf{q}|^2\bigr)^2 + \bigl(2\,\mathbf{p}\cdot\mathbf{q}\bigr)^2 = \bigl(|\mathbf{p}|^2+|\mathbf{q}|^2\bigr)^2 - \bigl|2\,\mathbf{p}\times\mathbf{q}\bigr|^2 = T^2-|\mathbf{X}|^2 , $$

the middle equality being the Lagrange identity $(\mathbf{p}\cdot\mathbf{q})^2+|\mathbf{p}\times\mathbf{q}|^2=|\mathbf{p}|^2|\mathbf{q}|^2$. This is the source's central formula, and it identifies $S^2=\sigma\sigma^*$ — an invariant of the algebra automorphisms, computed from the complex 3-vector — with a Minkowski interval in the coordinates $(T,\mathbf{X})$, which are bilinear in the complex coordinates of the primary space. The reality and the Lorentzian signature of the interval are not assumed anywhere: they are consequences of the bilinearity and of the identity.

The rotation and boost pieces. Under a real rotation of $x$ — the real subgroup of $SO(3,\mathbb{C})$ — the coordinates transform as

$$ T \longmapsto T , \qquad \mathbf{X} \longmapsto R\,\mathbf{X} , $$

a scalar and a 3-vector, which is the source's remark that $T$ is invariant under real 3-rotations. Under an imaginary rotation by $i\psi$ in the plane of a pair of components — the complementary, non-compact part of $SO(3,\mathbb{C})$ — with $z_3=0$ so that $\mathbf{X}$ lies along the third axis, one gets

$$ T \longmapsto \cosh 2\psi\,T + \sinh 2\psi\,X_3 , \qquad X_3 \longmapsto \sinh 2\psi\,T + \cosh 2\psi\,X_3 , $$

a Lorentz boost of rapidity $2\psi$. The doubling is the source's point: the Lorentz boost angle is twice the rotation angle that generates it, so the Lorentz action on the bilinear coordinates is not the same as the complex rotation that produces it. The two families together — real rotations and imaginary rotations — are the six real parameters of $SO(3,\mathbb{C})$, and their images are the rotations and the boosts of $SO^+(1,3)$, which is the isomorphism of the previous paragraph made explicit at the level of one-parameter subgroups.

Causality as a theorem. Two inequalities, both elementary, make the construction more than a rewriting. By AM–GM, $2|\mathbf{p}|\,|\mathbf{q}|\le|\mathbf{p}|^2+|\mathbf{q}|^2=T$; and $|\mathbf{X}|=2|\mathbf{p}\times\mathbf{q}|\le2|\mathbf{p}|\,|\mathbf{q}|$. Hence

$$ |\mathbf{X}| \;\le\; T , \qquad\text{that is,}\qquad S^2 = T^2-|\mathbf{X}|^2 \;\ge\; 0 . $$

The induced interval is timelike or null, never spacelike. The source draws the consequence: the real $(1+3)$ pseudo-Euclidean geometry and its causal structure — which special relativity has to postulate — are here consequences of the primary complex space. In this article's terms, the structural hypothesis that the Lorentzian metric is a consequence of the complex structure is not an interpretation laid on the algebra; on this route it is an identity, the Lagrange identity plus AM–GM. Equality $S=0$ holds exactly when the two inequalities are both saturated, that is when $|\mathbf{p}|=|\mathbf{q}|$ and $\mathbf{p}\cdot\mathbf{q}=0$: the complex null cone $\sigma=0$ maps to the real light cone, which is the source's infinitesimal statement of the same fact.

The geometric phase. The argument of the invariant $\sigma$ is itself invariant, being the argument of something the group fixes:

$$ \alpha = \arg\sigma , \qquad \tan\alpha = \frac{2\,\mathbf{p}\cdot\mathbf{q}}{|\mathbf{p}|^2-|\mathbf{q}|^2} , $$

the second form being the source's. So beside the Minkowski interval the construction yields a second Lorentz invariant, a phase. It is not a holonomy — there is no connection and no loop — but the coordinate that labels the fibre of the bilinear map: given $(T,\mathbf{X})$ the pair $(\mathbf{p},\mathbf{q})$ is determined up to a common rotation in the plane orthogonal to $\mathbf{X}$, and $\alpha$ is the remaining datum. The source calls it the geometric phase and proposes it as the origin of the quantum properties of particles, which its programme identifies with field singularities. That proposal is speculative and is recorded here as the source's, not as a result; what is a result is that $\alpha$ is Lorentz-invariant and computable from the complex 3-vector.

The source relates $\alpha$ to the motion by

$$ \cos^2\theta = \frac{1-v^2}{1+v^2\cot^2\alpha} , $$

where $\theta$ is the angle between $\mathbf{p}$ and $\mathbf{q}$ and $v=|\delta\mathbf{X}|/\delta T$ is the speed in units of $c$; at $v=0$ it forces $\cos^2\theta=1$, the pair parallel or antiparallel, which the source reads as a possible origin of spin. Both the invariance of $\alpha$ and this relation were checked on random data (below).

Two slips of the source, recorded rather than repaired. First, the power of $|z_0|$. The local complex null cone of the source's dynamics is $\sigma=z_0^2$ — its equation for the relative coordinate, equivalent to the vanishing of the determinant — so taking moduli gives $|\sigma|=|z_0|^2$, hence $S=|\sigma|=|z_0|^2$ and

$$ T^2-|\mathbf{X}|^2 = S^2 = |z_0|^4 . $$

The source prints $T^2-|\mathbf{X}|^2=S^2\equiv|z_0|^2$; the two differ by $|z_0|^2$, and the consistent reading is the first, since $\sigma=z_0^2$ is the nullness condition and is displayed correctly. Relatedly, $\alpha=\arg\sigma=2\arg z_0$, so the phase of the complex proper time is $\alpha/2$; the source's "the phase invariant corresponds to the phase of the complex proper time" is loose by that factor of two. Second, the letter $T$. The source defines $T=|\mathbf{p}|^2+|\mathbf{q}|^2$ as a coordinate and writes the interval element of a displacement as $\delta T=|\delta\mathbf{p}|^2+|\delta\mathbf{q}|^2$. The second is not the differential of the first: $dT=2(\mathbf{p}\cdot d\mathbf{p}+\mathbf{q}\cdot d\mathbf{q})$ is linear in the increments and integrates to zero around a closed loop, whereas $|\delta\mathbf{p}|^2+|\delta\mathbf{q}|^2$ is quadratic in them and positive. The source's claims that the induced time is irreversible and "non-holonomic" are therefore statements about the quadratic form — a metric, that is a Finsler-type arc element, not a differential — and a reader must not read the interval element as $dT$. Both points are checked below.

What the route does not give, and the boundary. The construction realizes the closed forward light cone only. There is no spacelike region, since $|\mathbf{X}|\le T$ identically, and no past cone, since $T\ge0$ identically; a spacelike separation or a past-directed causal relation cannot be represented at all. The map $(\mathbf{p},\mathbf{q})\mapsto(T,\mathbf{X})$ is onto that cone but is not a diffeomorphism — six real dimensions onto four — with fibre the phase and the common rotation, so the primary complex space is not Minkowski space in disguise. And the programme's own vocabulary for what the extra dimensions do — the "observable" space-time, the ensemble of correlated particle-singularities it calls duplicons, and the complex null cone whose defining equation has several roots — is the source's; the twistor-side content of it belongs to the companion article Twistor Theory and Biquaternions, and the vocabulary is not adopted here.

Verification. Every claim of this section was recomputed on 100 random instances. On random biquaternions and random $\tilde{M}\in SL(2,\mathbb{C})$: $z_0$, $\det\tilde{Q}$ and $\sigma$ were invariant under $\tilde{Q}\mapsto\tilde{M}\tilde{Q}\tilde{M}^{-1}$ to $1.6\times10^{-15}$, $1.1\times10^{-13}$ and $1.9\times10^{-13}$; the induced $3\times3$ matrix satisfied $R^TR=I$ to $6.0\times10^{-15}$ with $\det R=1$ to $8.4\times10^{-15}$. The Lagrange identity held to $2.3\times10^{-14}$, $|\mathbf{X}|\le T$ showed no violation, with equality to $8.9\times10^{-16}$ exactly when $|\mathbf{p}|=|\mathbf{q}|$ and $\mathbf{p}\cdot\mathbf{q}=0$. The imaginary rotation reproduced the boost of rapidity $2\psi$ to $1.6\times10^{-15}$ with $T^2-X_3^2$ invariant to $6.1\times10^{-14}$, and the real rotation rotated $\mathbf{X}$ and fixed $T$ to $3.3\times10^{-16}$. $T^2-|\mathbf{X}|^2$ and $\arg\sigma$ were invariant under random automorphisms to $7.1\times10^{-12}$ and $3.6\times10^{-14}$. The phase relation held to $4.4\times10^{-16}$. Every forward-cone datum $T\ge|\mathbf{X}|\ge0$ was attained ($100$ of $100$), and the common rotation about $\mathbf{X}$ fixed $T$, $|\mathbf{X}|$ and $\sigma$ to $8.0\times10^{-16}$. On closed polygonal loops in $(\mathbf{p},\mathbf{q})$, $T$ returned to its starting value to $1.3\times10^{-15}$ while the integral of the interval element was $1.8\times10^{-3}$ to $5.7\times10^{-3}$, positive as the second slip above predicts; and the source's nullness condition gave $T^2-|\mathbf{X}|^2=|z_0|^4$ to $5.8\times10^{-15}$.

The Infinitesimal Automorphisms: the Lie Algebra

Differentiating the unit-norm condition gives the Lie algebra of infinitesimal biquaternion-norm automorphisms. Let

$$ \tilde{\Lambda} = e_0 + \varepsilon\,\tilde{Q}, \qquad \varepsilon\in\mathbb{R},\quad \varepsilon\ll1 . $$

The condition $N(\tilde{\Lambda}) = 1$ becomes

$$ N(e_0+\varepsilon\tilde{Q}) = (e_0+\varepsilon\tilde{Q})(e_0+\varepsilon\tilde{Q}^{\natural}) = e_0 + \varepsilon\left(\tilde{Q}+\tilde{Q}^{\natural}\right) + O(\varepsilon^2) = e_0 + 2\varepsilon\,\mathrm{Sc}(\tilde{Q}) + O(\varepsilon^2), $$

so the tangent space at the identity is

$$ \mathrm{SL}(2,\mathbb{C}) = \{\tilde{Q}\in\mathbb{B} : \mathrm{Sc}(\tilde{Q}) = 0\}, $$

the six-real-dimensional space spanned by the three real units and the three imaginary units,

$$ \mathrm{SL}(2,\mathbb{C}) = \mathrm{span}_\mathbb{R}\{\,e_1,e_2,e_3,\ ie_1,ie_2,ie_3\,\}. $$

This is the Lie algebra of the automorphism group, and it splits into rotation generators $\mathcal{J}_k = e_k$ (real quaternion directions) and boost generators $\mathcal{K}_k = ie_k$ (imaginary vector directions). Their brackets are computed directly from the quaternion multiplication rule; the commutator is $[A,B] = AB-BA$, and

$$ [\mathcal{J}_j,\mathcal{J}_k] = 2\varepsilon_{jkl}\mathcal{J}_l, \qquad [\mathcal{J}_j,\mathcal{K}_k] = 2\varepsilon_{jkl}\mathcal{K}_l, \qquad [\mathcal{K}_j,\mathcal{K}_k] = -2\varepsilon_{jkl}\mathcal{J}_l . $$

The first two say that the rotations close and that the boosts transform as a vector under them; the third, with its minus sign, is the algebraic statement that two boosts do not close into a boost but into a rotation plus a boost. This is the infinitesimal form of the Thomas–Wigner rotation, and the sign is the one that makes the boost directions a vector and the rotation directions an axial vector.

Verification. The three bracket families above were checked exactly, as identities in the quaternion algebra, for all $j,k\in\{1,2,3\}$: they reproduce the standard Lorentz algebra $\mathrm{SO}(1,3)$ up to the conventional factor two. For example $[\mathcal{J}_1,\mathcal{J}_2] = 2e_3$ and $[\mathcal{K}_1,\mathcal{K}_2] = -2e_3$, both confirmed.

The complexification and the two factors. Complexifying the real Lie algebra and forming

$$ \mathcal{N}^{\pm}_k = \tfrac14\left(\mathcal{J}_k \pm \mathrm{i}\,\mathcal{K}_k\right), $$

where $\mathrm{i}$ is the complexification unit — distinct from the algebra's own central $i$ — gives two commuting copies of the rotation algebra,

$$ [\mathcal{N}^{+}_j,\mathcal{N}^{+}_k] = \varepsilon_{jkl}\mathcal{N}^{+}_l, \qquad [\mathcal{N}^{-}_j,\mathcal{N}^{-}_k] = \varepsilon_{jkl}\mathcal{N}^{-}_l, \qquad [\mathcal{N}^{+}_j,\mathcal{N}^{-}_k] = 0 . $$

The factor $\tfrac14$ is forced by the normalization $[\mathcal{J}_j,\mathcal{J}_k] = 2\varepsilon_{jkl}\mathcal{J}_l$. With $\mathcal{K}_k = ie_k$ and $\mathrm{i}$ a second central imaginary unit, the combination can be written $\mathcal{N}^{\pm}_k = \tfrac14\left(1 \pm \mathrm{i} i\right)e_k$, and its central factor is idempotent up to a factor two,

$$ \left(1 \pm \mathrm{i} i\right)^2 = 2\left(1 \pm \mathrm{i} i\right), \qquad\text{because}\qquad (\mathrm{i} i)^2 = \mathrm{i}^2 i^2 = +e_0 . $$

This is what makes each combination close on itself with the same structure constants, and what makes the two commute: $\left(1 + \mathrm{i} i\right)\left(1 - \mathrm{i} i\right) = 1 - (\mathrm{i} i)^2 = 0$. For $[\mathcal{N}^{+}_j,\mathcal{N}^{+}_k]$ the central factor contributes $(1+\mathrm{i} i)^2 = 2(1+\mathrm{i} i)$ and the quaternion commutator contributes $[e_j,e_k] = 2\varepsilon_{jkl}e_l$, so the product is $4\varepsilon_{jkl}\left(1+\mathrm{i} i\right)e_l/16 = \varepsilon_{jkl}\mathcal{N}^{+}_l$, as displayed. The combinations are the ones that diagonalize the adjoint action of the complexified boost generator. This is the Lie-algebra shadow of the group isomorphism $SO(4,\mathbb{C})\cong(SL(2,\mathbb{C})\times SL(2,\mathbb{C}))/\mathbb{Z}_2$ of the complex section: the two factors are the two commuting $\mathrm{SU}(2)$ algebras. The care needed here is that $\mathrm{i}$ is the complexification unit and not the algebra's scalar imaginary; the latter already appears in $\mathcal{K}_k = ie_k$, and conflating the two would be a notational error. The distinction is the same one that separates the real form $\mathrm{SO}(1,3)$ from its complexification.

What the Algebra Supplies and What Is Standard

Supplied by the algebra. The fact that the biquaternion norm is multiplicative, and therefore that it is the algebra's own quadratic structure; the determinant realization $N = \det$, which turns the automorphism problem into a problem about $M_2(\mathbb{C})$; the restriction of the complex form to the real slices, with the signature table; the identification of the automorphisms preserving the material slice with $O(1,3)$, and of the identity component with the rotor conjugations; and the infinitesimal algebra with its rotation-boost split and its minus sign on the boost-boost bracket.

Standard mathematics transcribed. The classification of nondegenerate complex quadratic forms, the isomorphism $SO(4,\mathbb{C})\cong(SL(2,\mathbb{C})\times SL(2,\mathbb{C}))/\mathbb{Z}_2$, the double cover $SL(2,\mathbb{C})\to SO^+(1,3)$, and the real forms of $\mathrm{SO}(4,\mathbb{C})$ are standard Lie theory. The surjectivity of the rotor map is the standard covering theorem and is cited, not re-derived; what is derived here is the algebraic form of the action and its kernel.

Interpretation. The reading of the material slice as physical spacetime, and of its biquaternion-norm automorphisms as the Lorentz group, is the framework's structural hypothesis. The group-theoretic content is exact; the physical assignment is the hypothesis, and it is the same hypothesis that the foundational articles The Anti-Hermitian Subspace $\mathbb{M}_-$ as the Material Sector and Introduction to the Biquaternion Universe state.

Attributed to a neighbouring programme. The diagonal reading of the algebra automorphisms, the bilinear real coordinates $T=|\mathbf{p}|^2+|\mathbf{q}|^2$, $\mathbf{X}=2\,\mathbf{p}\times\mathbf{q}$, the induced interval $T^2-|\mathbf{X}|^2=S^2=\sigma\sigma^*$ with its guaranteed sign, the boost of rapidity $2\psi$, and the invariant phase $\alpha=\arg\sigma$ are the construction of Kassandrov's algebrodynamics and not of this article. What is this article's is the placement of that construction as the middle case between the norm and the slice, the identification of its group with the diagonal $SO(3,\mathbb{C})\subset SO(4,\mathbb{C})$, and the checks. The source's reading of the phase as the origin of quantum properties, and its vocabulary of duplicons and "observable" space-time, are not adopted. The programme as a whole — its nonlinear primary equation, its twistor field, its multivalued principal field, its induced geometry with phase and its particle picture — is recorded, once, in The Algebrodynamical Programme: Nonlinear Cauchy–Riemann and Self-Quantized Charge, which owns it; this article's contribution is the group-theoretic identification of the automorphism group with the diagonal of the norm-automorphism group and the numerical checks.

Open Questions

1. Automorphisms of the full algebra. The maps considered here preserve the biquaternion norm and, at the real level, the material slice. The $\mathbb{C}$-algebra automorphisms of $\mathbb{B}\cong M_2(\mathbb{C})$ are the inner automorphisms, $\tilde{Q}\mapsto\tilde{A}\tilde{Q}\tilde{A}^{-1}$, a subgroup of the form automorphisms. The group-theoretic half of the question has a sharp answer, given in the section above: the subgroup is the diagonal $SO(3,\mathbb{C})\cong PGL(2,\mathbb{C})$ inside $SO(4,\mathbb{C})$, and as a real group it is the restricted Lorentz group itself — which is why a construction can take the algebra automorphisms, rather than the slice automorphisms, as the physical Lorentz group. What remains open is whether the framework adopts that reading, in which Minkowski space is a bilinear image of $\mathbb{C}^3$ and the causal structure follows from the Lagrange identity, or keeps the slice reading of the summary. The corpus records the construction and its checks, and leaves the choice to the foundational articles.

2. The discrete components. $O(1,3)$ has four components; the rotor group covers only $SO^+(1,3)$. Parity and time reversal are form automorphisms outside the rotor group. Whether the framework can represent them by an operation on biquaternion fields — rather than on four-vectors — without leaving the algebra is not settled here.

3. Real forms and the two sectors. Both $\mathbb{M}_-$ and $\mathbb{M}_+$ restrict the form to a Lorentzian signature — $(3,1)$ on the material slice and $(1,3)$ on the informational one — and $\mathbb{H}_{\mathbb{B}}$ to $(4,0)$. The Euclidean real form is thus available. Is the Euclidean form, and the compact group it defines, the home of the informational sector's own symmetries? The question connects this article to The Hermitian Subspace $\mathbb{M}_+$ as the Informational Sector.

4. The biquaternion norm and curved spacetime. The automorphism characterization is pointwise and flat. Whether it globalizes to a bundle of algebra automorphisms over a curved base, and what plays the role of the form there, is the same open question the foundational articles record for the whole framework.

5. Uniqueness of the physical group. The form determines $O(4,\mathbb{C})$ uniquely; the real structure then determines its real forms. Could a different real structure on $\mathbb{B}$ select a different physical group, and does the algebra rule out such a choice?

The conventions of the construction are those of the following companion articles:

• Companion article Introduction to the Biquaternion Universe, for the notation, the biquaternion norm and the sector structure.
• Companion article Conventions in the Biquaternion Universe, for the algebra and basis, the conjugations, the real subspaces and the metric at its three levels.
• Companion article The Lorentz Group in Biquaternionic Form — Structure and Representations, for the Lie algebra, the real forms and the representation theory.
• Companion article The Spinor Module in Biquaternionic Form and Its Lorentz Action, for the defining module and the action of the group on it.
• Companion article The Light Cone as the Biquaternion Zero-Divisor Cone, for the cone preserved by the automorphisms.
• Companion article The Two-Sheeted Cover and the Topology of Boosts in Biquaternionic Form, for the covering group and the global structure.

Summary

The biquaternion norm $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural} = \sum_\mu Q_\mu^2$ is a nondegenerate multiplicative quadratic form on $\mathbb{B}\cong\mathbb{C}^4$, with polarization matrix the identity and with

$$ N(\tilde{Q}\tilde{R}) = N(\tilde{Q})N(\tilde{R}), \qquad N(\tilde{Q}^{*}) = \overline{N(\tilde{Q})}. $$

Its automorphism group at the complex level is $O(4,\mathbb{C})$, whose connected component is

$$ SO(4,\mathbb{C}) \cong \frac{SL(2,\mathbb{C})\times SL(2,\mathbb{C})}{\{\pm(e_0,e_0)\}}, $$

realized by $\tilde{Q}\mapsto\tilde{A}\tilde{Q}\tilde{B}^{-1}$ with $\det\tilde{A}=\det\tilde{B}=1$. The algebra's real structure selects the anti-Hermitian slice $\mathbb{M}_-$, on which the form restricts to

$$ N(ict\,e_0+\mathbf{x}) = -c^2t^2+\mathbf{x}^2, \qquad \eta = \mathrm{diag}(-1,+1,+1,+1), $$

so the automorphisms preserving the slice form $O(1,3)$, with identity component the restricted Lorentz group. That component is exactly the group of rotor conjugations,

$$ SO^+(1,3)\cong SL(2,\mathbb{C})/\{\pm e_0\}, \qquad \tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}, \qquad N(\tilde{\Lambda})=1, $$

the map being two-to-one with kernel $\{\pm e_0\}$. Infinitesimally, the algebra is $\mathrm{SL}(2,\mathbb{C}) = \{X : \mathrm{Sc}(\tilde{Q})=0\}$, spanned by the rotation generators $\mathcal{J}_k = e_k$ and the boost generators $\mathcal{K}_k = ie_k$, with

$$ [\mathcal{J}_j,\mathcal{J}_k] = 2\varepsilon_{jkl}\mathcal{J}_l, \qquad [\mathcal{J}_j,\mathcal{K}_k] = 2\varepsilon_{jkl}\mathcal{K}_l, \qquad [\mathcal{K}_j,\mathcal{K}_k] = -2\varepsilon_{jkl}\mathcal{J}_l, $$

whose complexification splits into two commuting rotation algebras. The Lorentz group, in this reading, is what the biquaternion norm's automorphisms become when they are required to respect the algebra's real structure.

A second reading is available, and the two must not be conflated. The automorphisms that preserve the product and not only the form are the inner, or diagonal, ones,

$$ \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \cong PGL(2,\mathbb{C}) \cong PSL(2,\mathbb{C}) \cong SO(3,\mathbb{C}), \qquad SO(3,\mathbb{C}) \subset SO(4,\mathbb{C}) \ \text{the diagonal}, $$

and as a real group this is again the restricted Lorentz group. Acting on the complex 3-vector $z_a=p_a+iq_a$ it induces, through the bilinear real coordinates

$$ T=|\mathbf{p}|^2+|\mathbf{q}|^2 , \qquad \mathbf{X}=2\,\mathbf{p}\times\mathbf{q} , \qquad \sigma\sigma^* = \bigl(|\mathbf{p}|^2-|\mathbf{q}|^2\bigr)^2+\bigl(2\,\mathbf{p}\cdot\mathbf{q}\bigr)^2 = T^2-|\mathbf{X}|^2 , $$

where $\sigma=z_1^2+z_2^2+z_3^2$. On that route the Minkowski form is induced from the complex structure rather than read off a real slice, real rotations give the spatial rotations and imaginary rotations give boosts of twice the angle, and $T^2-|\mathbf{X}|^2\ge0$ is the Lagrange identity together with AM–GM, so the causal structure is a theorem rather than a postulate. The price is that only the closed forward cone is realized, that the primary space is complex 3-space with a Lorentz-invariant phase $\alpha=\arg\sigma$ rather than a real slice, and that the two routes therefore answer different questions and are recorded separately.

Summary of Notation

Further Reading

• Hermann Weyl, The Classical Groups: Their Invariants and Representations (Princeton, 1946), for the orthogonal groups and the isomorphisms of low-dimensional classical groups.
• Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press, 1978), for real forms of complex Lie algebras and the classification of the Lorentz group's real forms.
• Brian C. Hall, Lie Groups, Lie Algebras, and Representations (Springer, 2015), for the covering groups, the exponential map and the isomorphism $SO(4,\mathbb{C})\cong(SL_2\times SL_2)/\mathbb{Z}_2$.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for the identification of $SL(2,\mathbb{C})$ with the spin group and the double cover of the Lorentz group.
• Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1 (Cambridge, 1984), for the two-component realization of the restricted Lorentz group and its Lie algebra.
• Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists (Cambridge, 2003), for the rotor representation of the Lorentz group and the boost–rotation decomposition.
• David Hestenes, Space-Time Algebra (Gordon and Breach, 1966), for the even-subalgebra and bivector formulation of the Lorentz group.
• Steven Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge, 1995), for the Lorentz group, its complexification and its finite-dimensional representations.
• V. V. Kassandrov, "Algebrodynamics in Complex Space-Time and the Complex-Quaternionic Origin of Minkowski Geometry", arXiv:gr-qc/0602088 (2006), for the identification of the algebra automorphism group $SO(3,\mathbb{C})=PGL(2,\mathbb{C})$ as the proper Lorentz group acting on the bilinear real coordinates $T=|\mathbf{p}|^2+|\mathbf{q}|^2$, $\mathbf{X}=2\,\mathbf{p}\times\mathbf{q}$; for the identity $T^2-|\mathbf{X}|^2=\sigma\sigma^*\ge0$ that makes the causal structure a theorem; for the boost of rapidity $2\psi$ generated by an imaginary rotation; and for the Lorentz-invariant phase $\alpha=\arg\sigma$ and its relation to the speed. The same paper is cited, under its other title "On a quaternionic induced geometry with phase", in The Algebrodynamical Programme: Nonlinear Cauchy–Riemann and Self-Quantized Charge, which owns the programme and its particle picture. Two slips of the source are recorded in the section above: the power of $|z_0|$ in its (22), and the use of the letter $T$ for both the coordinate and the interval element of a displacement.