The Two-Sheeted Cover and the Topology of Boosts in Biquaternionic Form

Introduction

The preceding article identifies the restricted Lorentz group with the group of rotor conjugations of the biquaternion algebra,

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

and shows that the correspondence is two-to-one. The identification raises two questions that are not answered by the group law alone. What is the topology of the covering group, and what is the topology of the subgroup formed by the boosts? The two questions have contrasting answers, and the contrast is the subject of this article.

The covering group $SL(2,\mathbb{C})$ is simply connected. As a manifold it is diffeomorphic to $SU(2)\times\mathbb{R}^3$, and it deformation-retracts onto $SU(2)\cong S^3$. Consequently the map

$$ \pi:\ SL(2,\mathbb{C})\ \longrightarrow\ SO^+(1,3), \qquad \pi(\tilde{\Lambda}):\tilde{Q}\longmapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}, $$

is the universal cover of the restricted Lorentz group, and the fundamental group of that group is

$$ \pi_1\bigl(SO^+(1,3)\bigr) = \mathbb{Z}/2\mathbb{Z}. $$

The nontrivial class is generated by a rotation through $2\pi$: the rotor $R(\theta) = \cos(\theta/2) + \sin(\theta/2)\hat{\mathbf{n}}$ returns $R(2\pi) = -e_0$, not $e_0$, and the corresponding closed path in $SO^+(1,3)$ cannot be contracted. A rotation through $4\pi$ returns $e_0$ and is contractible. This is the algebraic origin of the difference between "double" and "single" valuedness, and it is the topology the biquaternion algebra makes available without any extra structure.

The boosts, by contrast, are topologically trivial. The pure boosts form a three-dimensional manifold diffeomorphic to $\mathbb{R}^3$ — the hyperbolic three-space carried by the positive Hermitian unit-norm elements — and every loop in it is contractible. A boost can never be deformed continuously into a rotation while remaining a boost, and the obstruction to closing two boosts into a boost is carried entirely by the rotation that their product contains. That rotation is the Thomas–Wigner rotation, and its angle is computed here in the rotor algebra.

The article develops three threads. The first is the manifold structure of the covering group and its contraction onto $SU(2)$. The second is the covering itself, the $2\pi$ loop and the belt picture. The third is the topology of the boost manifold, the non-closure of the boost composition law, and the Wigner angle. The closing sections separate the algebraic content from the standard topology and record the open questions.

Boundaries. The physical consequences of the double cover for spinning matter — spinor representations, the transformation law of a spinor under a $2\pi$ rotation, and the observed sign change — belong to the sibling category on relativistic quantum theory and are not developed here. This article is the group-theoretic and geometric statement: what the covering group is, what its fundamental group is, and what the boost manifold is. The structure and representations of the group are treated in the companion article The Lorentz Group in Biquaternionic Form — Structure and Representations, and the automorphism characterization in The Lorentz Group as Biquaternion Norm Automorphisms.

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 and real vector — the material sector), $\mathbb{M}_+$ (Hermitian: real scalar and imaginary vector — the informational sector), $\mathbb{H}_{\mathbb{B}}$ (real quaternions) and $\mathbb{C}_{\mathbb{B}}$ (the center). The conjugations are ${}^{\natural}$, $\bar{\cdot}$, ${}^{*} = ({}^{\natural})^{\,*}$ and ${}^\flat = -{}^{*}$. The biquaternion norm is $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural} = \sum_\mu Q_\mu^2$, the level-1 identity form $\mathrm{diag}(+1,+1,+1,+1)$ on $\mathbb{C}$, restricting to the level-2 form $\eta = \mathrm{diag}(-1,+1,+1,+1)$ on $\mathbb{M}_-$. A rotor is a unit-norm biquaternion, $N(\tilde{\Lambda}) = e_0$, and a general rotor decomposes as

$$ \tilde{\Lambda} = R\,B, \qquad R\in SU(2)\subset\mathbb{H}_{\mathbb{B}}, \qquad B\ \text{a pure boost}, $$

with $R$ a spatial rotation and $B$ a boost. A pure boost of rapidity $\psi$ along the unit real vector $\hat{\mathbf{u}}$ is

$$ B(\psi,\hat{\mathbf{u}}) = \cosh\frac{\psi}{2} + i\,\sinh\frac{\psi}{2}\,\hat{\mathbf{u}}, \qquad B\in\mathbb{M}_+, \qquad B = B^\dagger . $$

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 Manifold of the Covering Group

The unit-norm elements

$$ SL(2,\mathbb{C}) = \{\tilde{\Lambda}\in\mathbb{B} : N(\tilde{\Lambda}) = e_0\} $$

form a real Lie group of dimension six, because the single complex equation $N(\tilde{\Lambda}) = 1$ is two real conditions on the eight real coordinates of $\tilde{\Lambda}$. It is the double cover of the restricted Lorentz group and, as a manifold, it has a simple structure given by the polar decomposition.

Every invertible matrix has a unique decomposition into a unitary and a positive Hermitian factor; transported to the algebra, this reads

$$ \tilde{\Lambda} = R\,B, \qquad R^{*} = R^{-1}, \qquad B^\dagger = B, \qquad B\ \text{positive} . $$

For a unit-norm rotor the two factors are themselves unit-norm: $N(R) = N(B) = 1$. The unitary factor lives in the compact subgroup

$$ SU(2) = \{\,R\in\mathbb{H}_{\mathbb{B}} : N(R) = e_0\,\}\cong S^3, $$

which is the unit sphere in the four-dimensional real-quaternion space and therefore diffeomorphic to the three-sphere. The positive Hermitian factor is a boost matrix, and we will see that the boosts form a space diffeomorphic to $\mathbb{R}^3$. The polar decomposition therefore gives a diffeomorphism

$$ SL(2,\mathbb{C}) \;\cong\; SU(2)\times\mathbb{R}^3 , $$

and hence a deformation retraction

$$ SL(2,\mathbb{C}) \;\simeq\; SU(2)\cong S^3 . $$

Consequences. Since $S^3$ is simply connected and has $\pi_3(S^3) = \mathbb{Z}$, the covering group is simply connected and

$$ \pi_1\bigl(SL(2,\mathbb{C})\bigr) = 0, \qquad \pi_3\bigl(SL(2,\mathbb{C})\bigr) = \mathbb{Z}. $$

A simply connected cover of a connected group is its universal cover, so the two-to-one map $\pi$ of the introduction is the universal covering map of $SO^+(1,3)$. The whole topology of the covering group is the topology of $SU(2)$: the non-compact boost directions are contractible and contribute nothing to the homotopy.

Verification. The polar decomposition was checked numerically on random unit-norm rotors. Writing $\tilde{\Lambda} = RB$, the positive factor is the left factor of the pair,

$$ \tilde{\Lambda}^{*}\tilde{\Lambda} = B^\dagger R^{*} R B = B\,R^{*} R\,B = B^2, \qquad B = (\tilde{\Lambda}^{*}\tilde{\Lambda})^{1/2}, $$

using $R^{*} R = e_0$; the other ordering gives $\tilde{\Lambda}\tilde{\Lambda}^{*} = RB^2R^{-1}$, the conjugated boost $RBR^\dagger$, which is not the boost factor of $\tilde{\Lambda}$. With $R = \tilde{\Lambda}B^{-1}$ the factor $R$ was unitary and lay in $\mathbb{H}_{\mathbb{B}}$, the factor $B$ was Hermitian positive with $N(B)=e_0$, and the reconstruction $RB$ reproduced $\tilde{\Lambda}$ to machine precision. The eigenvalues of $B$ were real and positive, as required for the positive square root.

The Two-Sheeted Cover

The conjugation map

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

is a continuous surjective homomorphism from $SL(2,\mathbb{C})$ onto the restricted Lorentz group, and its kernel is

$$ \ker\pi = \{\pm e_0\}\cong\mathbb{Z}/2\mathbb{Z}, $$

as established by the automorphism article: $\pm e_0$ act identically on $\mathbb{M}_-$ and no other unit-norm element does. Hence

$$ SO^+(1,3)\cong SL(2,\mathbb{C})/\{\pm e_0\}, $$

and $\pi$ is a two-sheeted covering map: over every Lorentz transformation lie exactly two rotors, differing by the sign $-e_0$. Since the total space is simply connected, this is the universal cover, and the fundamental group of the base is the kernel of the covering,

$$ \pi_1\bigl(SO^+(1,3)\bigr)\cong\ker\pi\cong\mathbb{Z}/2\mathbb{Z}. $$

The statement is worth making twice, because it is the algebraic heart of the matter: the restricted Lorentz group has exactly one nontrivial loop, and it is the image of a path in $SL(2,\mathbb{C})$ that joins $e_0$ to $-e_0$. The rotor group itself is simply connected, so nothing is lost by working with rotors; what is lost by working with four-vectors is precisely the distinction between the two sheets.

The $2\pi$ Loop and the Belt

The generating loop is explicit. Let $\hat{\mathbf{n}}$ be any unit real vector and define the rotation rotor

$$ R(\theta) = \cos\frac{\theta}{2} + \sin\frac{\theta}{2}\,\hat{\mathbf{n}}\,\in\mathbb{H}_{\mathbb{B}}, \qquad N(R(\theta)) = \cos^2\frac{\theta}{2} + \sin^2\frac{\theta}{2} = 1 . $$

At $\theta = 0$ the rotor is $e_0$; at $\theta = 2\pi$,

$$ R(2\pi) = \cos\pi + \sin\pi\,\hat{\mathbf{n}} = -e_0, \qquad R(4\pi) = \cos 2\pi + \sin 2\pi\,\hat{\mathbf{n}} = e_0 . $$

The half-angle dependence is the whole phenomenon. The path

$$ \theta\ \longmapsto\ \pi\bigl(R(\theta)\bigr)\in SO^+(1,3), \qquad 0\leq\theta\leq 2\pi, $$

is a closed loop in the Lorentz group, because $\pi(R(2\pi)) = \pi(-e_0) = \pi(e_0)$. It is not null-homotopic: if it were, the path $R(\theta)$ in the simply connected group $SL(2,\mathbb{C})$ could be deformed, with endpoints fixed, to the constant path at $e_0$, and its endpoint $R(2\pi) = -e_0$ would equal $e_0$, a contradiction. The loop for $0\leq\theta\leq 4\pi$ is null-homotopic, because $R(4\pi) = e_0$ and $SL(2,\mathbb{C})$ is simply connected. Thus the generator of $\pi_1$ is a path of rotations of total angle $2\pi$, traversed twice to become contractible.

This is the familiar belt or plate trick: a rotation by $2\pi$ of an object attached to its surroundings by strands cannot be undone while the strands are held fixed, whereas a rotation by $4\pi$ can. Here the strands have an algebraic meaning — they are the continuous choice of which rotor represents the frame — and the belt trick is the statement that the continuous choice, followed around a $2\pi$ rotation, returns the negative of the starting rotor. The two endpoints $e_0$ and $-e_0$ are the same Lorentz transformation and different rotors.

Verification. The rotor path was evaluated numerically: $R(2\pi) = -e_0$ and $R(4\pi) = e_0$ were confirmed exactly, and the conjugation action of $R(\theta)$ on a fixed four-vector was periodic in $\theta$ with period $2\pi$, while the rotor itself was periodic only with period $4\pi$. The action is thus single-valued on four-vectors and the rotor is double-valued, which is the covering statement in coordinates.

The Topology of the Boost Manifold

The boosts are the other factor of the polar decomposition, and their manifold is simpler. A pure boost of rapidity $\psi$ along the unit direction $\hat{\mathbf{u}}$ has the rotor

$$ B(\psi,\hat{\mathbf{u}}) = \cosh\frac{\psi}{2} + i\,\sinh\frac{\psi}{2}\,\hat{\mathbf{u}} = \exp\!\left(\frac{\psi}{2}\,i\hat{\mathbf{u}}\right), \qquad \psi\in\mathbb{R}, \qquad \hat{\mathbf{u}}\in S^2 . $$

It is Hermitian, $B^\dagger = B$, of unit norm, and positive. The pair $(\psi,\hat{\mathbf{u}})$ is not unique: $\psi = 0$ is the identity for every direction, and $(\psi,\hat{\mathbf{u}})$ and $(-\psi,-\hat{\mathbf{u}})$ describe the same boost. The boost manifold is therefore the space of traceless Hermitian generators,

$$ \mathcal{B} = \left\{\, \tfrac{\psi}{2}\,i\hat{\mathbf{u}} : \psi\in\mathbb{R},\ \hat{\mathbf{u}}\in S^2 \,\right\} = i\,\mathbb{R}^3 , $$

the image of the exponential map on the three-dimensional space of pure imaginary vectors, and it is diffeomorphic to $\mathbb{R}^3$ with coordinates $\boldsymbol{\beta} = \psi\,\hat{\mathbf{u}}$. It is contractible: every loop in $\mathcal{B}$ shrinks to a point, and there is no nontrivial topology to be found.

The exponential parametrization is global and injective from $\mathbb{R}^3$ onto the boosts, because for a pure imaginary vector $\mathbf{w} = i\psi\hat{\mathbf{u}}$ the exponential is the closed form above, and

$$ B(\psi,\hat{\mathbf{u}})\,B(\psi',\hat{\mathbf{u}}) = B(\psi+\psi',\hat{\mathbf{u}}) $$

for collinear directions; the rapidity is the additive coordinate on each line. The boosts therefore form a symmetric space

$$ \mathcal{B}\cong SL(2,\mathbb{C})/SU(2)\cong\mathbb{H}^3, $$

hyperbolic three-space: the coset of the covering group by its maximal compact subgroup. The rotation group acts on it by conjugation, $B\mapsto RBR^\dagger$, and this action is transitive on directions. Hyperbolic three-space is simply connected and contractible; the boost manifold inherits both properties.

The topology lives in the rotations. The polar decomposition and the coset description combine into the fiber bundle

$$ SU(2)\ \longrightarrow\ SL(2,\mathbb{C})\ \longrightarrow\ SL(2,\mathbb{C})/SU(2)\cong\mathbb{H}^3 , $$

or, passing to the quotient by $\{\pm e_0\}$,

$$ SO(3)\ \longrightarrow\ SO^+(1,3)\ \longrightarrow\ \mathbb{H}^3 . $$

In either form the base is contractible, so the total space is homotopy-equivalent to the fiber. The fundamental group of the Lorentz group is therefore the fundamental group of the rotation group,

$$ \pi_1\bigl(SO^+(1,3)\bigr)\cong\pi_1\bigl(SO(3)\bigr)\cong\mathbb{Z}/2\mathbb{Z}, $$

and its third homotopy group is $\pi_3(S^3) = \mathbb{Z}$, carried by the compact fiber. The non-compact boost directions contribute nothing to the topology; the entire $\mathbb{Z}/2$ is the rotation group's. A boost path is always contractible within the boosts, so no boost by itself can represent the nontrivial class.

Boosts Do Not Close: the Wigner Angle

Boosts do not form a group. The product of two boost rotors is a unit-norm rotor, hence by the polar decomposition a rotation times a boost,

$$ B(\psi_1,\hat{\mathbf{u}}_1)\,B(\psi_2,\hat{\mathbf{u}}_2) = R(\omega,\hat{\mathbf{h}})\,B(\psi_{\mathrm{eff}},\hat{\mathbf{u}}_{\mathrm{eff}}), $$

and the rotation factor is present whenever the two boost directions are not collinear. This rotation is the Thomas–Wigner rotation; it is the group-theoretic statement that the boost manifold is not a subgroup, and it is the physical face of the fact recorded above, that the topology and the non-closure both come from the rotation directions.

For two boosts of equal rapidity $\psi$ whose directions make an angle $\theta$, the Wigner rotation axis is the normal to the plane of the two directions and the Wigner angle $\omega$ is given by

$$ \boxed{\; \tan\frac{\omega}{2} = \frac{\sinh^2\dfrac{\psi}{2}\,\sin\theta} {\cosh^2\dfrac{\psi}{2} + \sinh^2\dfrac{\psi}{2}\,\cos\theta} \;} $$

This is the equal-rapidity form of the standard composition law; the general case of unequal rapidities is obtained by the same polar decomposition with two different rapidities.

Derivation. Take the first boost along $\hat{\mathbf{u}}_1 = \hat{\mathbf{z}}$ and the second along $\hat{\mathbf{u}}_2 = \sin\theta\,\hat{\mathbf{x}} + \cos\theta\,\hat{\mathbf{z}}$, so that the plane is spanned by $\hat{\mathbf{x}}$ and $\hat{\mathbf{z}}$ and the normal is $\hat{\mathbf{y}}$. Writing $c = \cosh(\psi/2)$, $s = \sinh(\psi/2)$, the product has the expansion

$$ B_1B_2 = \left(c + is\,e_3\right)\left(c + is\left(\sin\theta\,e_1 + \cos\theta\,e_3\right)\right). $$

Using $e_3e_1 = e_2$ and $e_3^2 = -e_0$, the product is

$$ B_1B_2 = \left(c^2 + s^2\cos\theta\right) + i\,sc\sin\theta\,e_1 - s^2\sin\theta\,e_2 + i\,sc\left(1+\cos\theta\right)e_3 . $$

The rotation part of this product is a rotation about $e_2$, the normal to the plane, so write $R = \cos\frac{\omega}{2} + \sin\frac{\omega}{2}\hat{\mathbf{h}}$ with $\hat{\mathbf{h}} = \pm e_2$. The configuration is symmetric under the reflection that fixes the $e_1$–$e_3$ plane and reverses $e_2$, so $B$ carries no $e_2$ component and $R$ lies along $e_0$ and $e_2$; the numerical polar decomposition below confirms both. Writing $B = \cosh\frac{\psi'}{2} + i\sinh\frac{\psi'}{2}\hat{\mathbf{m}}$ with $\hat{\mathbf{m}}$ in the $e_1$–$e_3$ plane, and writing $M_k$ for the coefficient of $e_k$ in a biquaternion $M$, the scalar and $e_2$ components of $L = RB$ are therefore

$$ \mathrm{Sc}(L) = \cos\frac{\omega}{2}\,\cosh\frac{\psi'}{2}, \qquad L_2 = \cosh\frac{\psi'}{2}\,R_2 , $$

because the vector parts of $R$ (along $e_2$) and of $B$ (in the $e_1$–$e_3$ plane) are orthogonal, and because the product of $e_2$ with the $e_1$–$e_3$ part of $B$ lies in the $e_1$–$e_3$ plane. Comparing with the expansion above, whose scalar part is $c^2+s^2\cos\theta$ and whose $e_2$ coefficient is $-s^2\sin\theta$, the common factor $\cosh\frac{\psi'}{2}$ cancels in the ratio and

$$ \tan\frac{\omega}{2} = \frac{s^2\sin\theta}{c^2 + s^2\cos\theta}, $$

which is the displayed formula in terms of $c = \cosh(\psi/2)$ and $s = \sinh(\psi/2)$. The verification below confirms it numerically.

Verification. The formula was checked against a direct numerical polar decomposition of the product of two unit-norm boost rotors at five parameter pairs. For $\psi = 1.0$, $\theta = 0.7$ the computed and formula values of $\omega$ agreed as $0.2354236579$ versus $0.2354236579$; the same agreement to ten digits was found at $(\psi,\theta) = (0.5,1.2)$, $(2.0,2.0)$, $(0.2,0.3)$ and $(3.0,0.9)$. The axis was confirmed to be the normal to the boost plane in each case.

Limits. The formula has two instructive limits. For small rapidity, $\sinh(\psi/2)\approx\psi/2$ and $\cosh(\psi/2)\approx1$, so

$$ \omega\approx\frac{\psi^2}{2}\sin\theta \qquad(\psi\ll1), $$

a purely relativistic effect: it vanishes as $v^2/c^2$. For large rapidity, $\sinh(\psi/2)\approx\cosh(\psi/2)\approx e^{\psi/2}/2$, so the ratio tends to $\sin\theta/(1+\cos\theta) = \tan(\theta/2)$ and

$$ \omega\ \longrightarrow\ \theta \qquad(\psi\to\infty), $$

the Wigner angle saturating at the angle between the two boost directions. The verification confirmed both limits: at $\psi = 20$ and $\theta = 1.0$ the formula gave $\omega = 0.9999999955$, and at $\psi = 0.001$ it gave $\omega = 4.2\times10^{-7}$.

The interpretation of the limit is worth stating, because it is purely kinematic and non-quantum. A single boost is a boost; two boosts in different directions are a boost accompanied by a rotation whose angle grows with the rapidity from zero and saturates at the angle between the directions. The composition of frame transformations is therefore path-dependent in the boost directions, and the rotation that closes the gap is the holonomy of the boost manifold. It is the same rotation that generates the nontrivial loop of the previous sections, now read as the obstruction to boosts forming a subgroup.

What the Topology Says

Three statements summarize the topology, and each is exact and free of quantum content.

The cover is two-to-one and the total space is simply connected. $SL(2,\mathbb{C})$ is the universal cover of $SO^+(1,3)$, the kernel is $\{\pm e_0\}$, and $\pi_1(SO^+(1,3)) = \mathbb{Z}/2$. The single nontrivial loop is a rotation by $2\pi$.

The boosts are contractible and carry no topology. The boost manifold is $\mathbb{R}^3\cong\mathbb{H}^3$, the symmetric space $SL(2,\mathbb{C})/SU(2)$. Every loop of boosts is contractible; the whole topology of the group is the rotation group's.

The composition of boosts is obstructed by a rotation. Two non-collinear boosts compose to a boost times a Wigner rotation, with the exact angle above. The obstruction is measured by the same compact subgroup that carries the topology.

It is worth being precise about what is not claimed here. The sign that a rotor acquires on a $2\pi$ rotation is invisible on every element of the material sector: $\pi(R(2\pi)) = \pi(e_0)$, so no four-vector distinguishes the two sheets. The distinction becomes visible only for objects on which the algebra acts one-sidedly rather than by conjugation — the spinor module — and that is the entrance to the relativistic quantum theory, treated in the sibling category and in the companion article The Spinor Module in Biquaternionic Form and Its Lorentz Action. Within the non-quantum theory the double cover is a statement about the group manifold, and it has no observable consequence for four-vectors, fields or the stress–energy tensor.

What the Algebra Supplies and What Is Standard

Supplied by the algebra. The explicit rotor parametrization of the cover, and therefore the concrete loop $R(\theta) = \cos(\theta/2)+\sin(\theta/2)\hat{\mathbf{n}}$ with $R(2\pi) = -e_0$; the identification of the two sheets with the sign $\pm e_0$; the boost rotor $B = \exp(\tfrac{\psi}{2}i\hat{\mathbf{u}})$ and the global exponential parametrization of the boost manifold by $\mathbb{R}^3$; and the derivation of the Wigner angle and its limits directly from the quaternion product and the polar decomposition.

Standard topology transcribed. The polar decomposition of $GL_2(\mathbb{C})$, the diffeomorphism $SL(2,\mathbb{C})\cong SU(2)\times\mathbb{R}^3$, the deformation retraction, the fibration $SO(3)\to SO^+(1,3)\to\mathbb{H}^3$, and the homotopy groups $\pi_1(SO(3)) = \mathbb{Z}/2$, $\pi_3(S^3) = \mathbb{Z}$ are standard Lie-group topology. The Thomas–Wigner rotation and its angle are standard relativistic kinematics. The algebra reproduces and unifies them; it does not replace them.

Interpretation. The reading of the two sheets as the algebraic home of the spin degree of freedom is the framework's hypothesis and belongs to the quantum sibling; the non-quantum reading is simply that the rotor, not the four-vector, is the element that carries the frame, and that the rotor remembers a $2\pi$ rotation while the four-vector does not.

Open Questions

The full Lorentz group and its cover. The two-to-one cover established here is that of the identity component $SO^+(1,3)$. The full group $O(1,3)$ has four components, and its covering groups are extensions involving the discrete transformations. Whether the biquaternion algebra supplies a single algebraic object covering all four components, and how parity and time reversal act on rotors, is not settled by the conjugation action alone.
The Euclidean real form and its topology. The same algebra contains the Euclidean real form on $\mathbb{H}_{\mathbb{B}}$, whose group is compact and whose topology is different. The relation between the two real forms' topologies, and whether the informational sector prefers the compact one, is an open question connecting this article to The Lorentz Group as Biquaternion Norm Automorphisms.
Gravity and the boost manifold. The boost manifold is hyperbolic three-space, which is also the spatial slice of a cosmological or hyperbolic geometry. Whether the framework's boost space has any relation to a physical spatial geometry, or merely shares the homogeneous-space structure, is not addressed here.
The Wigner angle at the level of fields. For a field on $\mathbb{M}_-$ the Wigner rotation acts by rotor conjugation like any other Lorentz transformation. Does the composition of two boosts, with its rotation factor, have an observable consequence for a classical field configuration beyond the standard relativistic one? This question shades into the spin-direction effects treated in the sibling subcategories and is not pursued here.
The $2\pi$ sign and the sector structure. The sign acquired by a rotor under a $2\pi$ rotation is invisible on $\mathbb{M}_-$ and would be visible on an object acted on one-sidedly. A non-quantum classical object with that property is not exhibited in this article; whether one exists is open.

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 rotor parametrisation.
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 as Biquaternion Norm Automorphisms, for the group as an algebraic object and its Lie algebra.
Companion article The Lorentz Group in Biquaternionic Form — Structure and Representations, for the representation theory of the group and its real forms.
Companion article The Spinor Module in Biquaternionic Form and Its Lorentz Action, for the module on which the two-to-one sign is seen.

Summary

The covering group of the restricted Lorentz group is the group of unit-norm biquaternions,

$$ SL(2,\mathbb{C}) = \{\tilde{\Lambda} : N(\tilde{\Lambda}) = e_0\} \cong SU(2)\times\mathbb{R}^3 \simeq S^3 , $$

which is simply connected, with $\pi_1 = 0$ and $\pi_3 = \mathbb{Z}$. The conjugation map $\pi(\tilde{\Lambda}):\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$ has kernel $\{\pm e_0\}$ and is the universal cover,

$$ SO^+(1,3)\cong SL(2,\mathbb{C})/\{\pm e_0\}, \qquad \pi_1\bigl(SO^+(1,3)\bigr) = \mathbb{Z}/2\mathbb{Z}. $$

The generator is the rotation rotor $R(\theta) = \cos(\theta/2)+\sin(\theta/2)\hat{\mathbf{n}}$, with $R(2\pi) = -e_0$ and $R(4\pi) = e_0$; the corresponding closed path in the Lorentz group is contractible only after being traversed twice. This is the belt picture, and it is the algebraic statement that the rotor, not the four-vector, carries the frame.

The boost manifold is

$$ \mathcal{B} = \left\{\,B(\psi,\hat{\mathbf{u}}) = \cosh\tfrac{\psi}{2} + i\sinh\tfrac{\psi}{2}\hat{\mathbf{u}}\,\right\} \cong\mathbb{R}^3\cong\mathbb{H}^3\cong SL(2,\mathbb{C})/SU(2), $$

contractible and carrying none of the topology; the decomposition of the group into rotations and boosts is the fibration $SO(3)\to SO^+(1,3)\to\mathbb{H}^3$, whose contractible base leaves the entire $\mathbb{Z}/2$ in the rotation fiber. Boosts do not form a subgroup: for two equal-rapidity boosts separated by $\theta$,

$$ B_1B_2 = R(\omega,\hat{\mathbf{h}})\,B_{\mathrm{eff}}, \qquad \tan\frac{\omega}{2} = \frac{\sinh^2\tfrac{\psi}{2}\sin\theta}{\cosh^2\tfrac{\psi}{2}+\sinh^2\tfrac{\psi}{2}\cos\theta}, $$

with $\omega\approx\tfrac{\psi^2}{2}\sin\theta$ for small rapidity and $\omega\to\theta$ as $\psi\to\infty$. The two-sheeted cover and the Wigner rotation are the same compact structure seen from the topological and the compositional side.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra, $\cong M_2(\mathbb{C})$
$N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural} = \sum_\mu Q_\mu^2$	Biquaternion norm; level-1 identity on $\mathbb{C}$
$\mathbb{M}_-, \mathbb{M}_+$	Material (anti-Hermitian) and informational (Hermitian) sectors
$\mathbb{H}_{\mathbb{B}}, \mathbb{C}_{\mathbb{B}}$	Real-quaternion subspace; complex scalar line (center)
$\tilde{\Lambda}$, $N(\tilde{\Lambda}) = e_0$	Unit-norm biquaternion; Lorentz rotor
$\pi(\tilde{\Lambda}):\tilde{Q}\mapsto\tilde{\Lambda}\tilde{Q}\tilde{\Lambda}^{*}$	Conjugation action on $\mathbb{M}_-$
$\ker\pi = \{\pm e_0\}$, $\pi_1(SO^+(1,3))=\mathbb{Z}/2$	Two-sheeted universal cover
$SL(2,\mathbb{C}) = \{\tilde{\Lambda}:N(\tilde{\Lambda})=e_0\}\cong SU(2)\times\mathbb{R}^3$	Covering group manifold
$R(\theta) = \cos\frac{\theta}{2} + \sin\frac{\theta}{2}\hat{\mathbf{n}}$	Rotation rotor; $R(2\pi)=-e_0$, $R(4\pi)=e_0$
$B(\psi,\hat{\mathbf{u}}) = \cosh\frac{\psi}{2} + i\sinh\frac{\psi}{2}\hat{\mathbf{u}}$	Pure boost rotor, in $\mathbb{M}_+$
$\mathcal{B}\cong\mathbb{R}^3\cong\mathbb{H}^3\cong SL(2,\mathbb{C})/SU(2)$	Boost manifold (hyperbolic three-space), contractible
$SO(3)\to SO^+(1,3)\to\mathbb{H}^3$	Fibration; topology carried by the rotation fiber
$\tan\frac{\omega}{2} = \frac{\sinh^2(\psi/2)\sin\theta}{\cosh^2(\psi/2)+\sinh^2(\psi/2)\cos\theta}$	Wigner angle for two equal-rapidity boosts
$\omega\to\frac{\psi^2}{2}\sin\theta$ ($\psi\ll1$), $\omega\to\theta$ ($\psi\to\infty$)	Small- and large-rapidity limits
$c = 1/\sqrt{\epsilon\mu}$, $c_0$	Speed of light in the medium; in vacuum