Split-Quaternion Rotations and the Lorentz Group

Introduction

This article studies the action of the unit split-quaternions on the vector subspace. It identifies the unit groups, computes the adjoint action, proves the double cover $\mathrm{SL}_2(\mathbb{R}) \to \mathrm{SO}^{+}(2,1)$ of the Lorentz group of signature $(2,1)$, describes the elliptic and hyperbolic one-parameter subgroups, and classifies the elements of the vector subspace as timelike, lightlike or spacelike according to their orbits.

The split-quaternion algebra, its vector subspace $V$, its split-quaternion norm $N$, its conjugation and its idempotents are assumed from Split-Quaternion Algebra; the group of units, the norm-one group and the signature of the restricted form are assumed from Split-Quaternion Norm and Invertibility. The Lorentz groups and the orthogonal groups are those of The Orthogonal Lie Algebra and Isometries and Orthogonal Transformations; the hyperbolic plane that the sheets carry is the subject of Split-Quaternions and Hyperbolic Geometry, to which the last step of this article points. Nothing physical is invoked.

The Unit Split-Quaternions

Definition. The norm-one group of the algebra is

$$ U = \{\tilde q \in \mathbb{H}_{\mathrm{s}} : N(\tilde q) = 1\}, $$

and the set of elements of norm $\pm 1$ is

$$ U^{\pm} = \{\tilde q \in \mathbb{H}_{\mathrm{s}} : N(\tilde q) = \pm 1\} = \mathrm{SL}_2^{\pm}(\mathbb{R}). $$

Both are groups under multiplication, and $U$ is a subgroup of $U^{\pm}$.

Theorem (The Two Components). The group $U$ is isomorphic to $\mathrm{SL}_2(\mathbb{R})$ and is connected; the group $U^{\pm}$ has exactly two connected components,

$$ U = \mathrm{SL}_2(\mathbb{R}) \quad \text{and} \quad \{\tilde q : N(\tilde q) = -1\} = \mathrm{SL}_2^{-}(\mathbb{R}), $$

and the two components are interchanged by multiplication by $e_2$. The full group of units is

$$ \mathbb{H}_{\mathrm{s}}^{\times} = \{N \neq 0\}, $$

its centre is $\mathbb{R}^{\times}$, and it has the two components $\{N > 0\}$ and $\{N < 0\}$.

Proof. The identification $U \cong \mathrm{SL}_2(\mathbb{R})$ and the description of the unit group are (Split-Quaternion Norm and Invertibility, §The Group of Units), where the connectedness of the norm-one group and the splitting of the units into the two components $\{N>0\}$, $\{N<0\}$ are recorded. The element $e_2$ has $N(e_2) = -1$ and $e_2^{-1} = e_2$, so multiplication by $e_2$ exchanges the two split-quaternion norm levels; it therefore identifies the two components of $U^{\pm}$, and since $U$ is connected, $U^{\pm}$ has exactly two components. The centre is $\mathbb{R}$ by (Split-Quaternion Algebra, §The Centre and Simplicity), so the central units are the nonzero scalars.

Remark (Two readings of "the unit split-quaternions"). The norm-one group $U$ is connected and has no two-component structure; the two components appear for the group $U^{\pm}$ of units of norm $\pm 1$. Both readings occur in the literature, and this article keeps them apart: the double cover below is a statement about $U = \mathrm{SL}_2(\mathbb{R})$, and the second component of $U^{\pm}$ is reached by $e_2$ and acts by isometries reversing the time direction.

The Adjoint Action on the Vector Subspace

For a unit $u$ define the conjugation map

$$ \Theta(u) : V \to V, \qquad \Theta(u)v = u v u^{-1}. $$

Theorem (The Adjoint Representation). For every unit $u$, the map $\Theta(u)$ is a real-linear automorphism of $V$ preserving the form $N$; the assignment $u \mapsto \Theta(u)$ is a group homomorphism

$$ \Theta : \mathbb{H}_{\mathrm{s}}^{\times} \longrightarrow O(V, N) \cong O(2,1), $$

whose kernel is the centre $\mathbb{R}^{\times}$ and whose image lies in $SO(2,1)$. On the norm-one group the restriction

$$ \Theta : U \cong \mathrm{SL}_2(\mathbb{R}) \longrightarrow \mathrm{SO}^{+}(2,1) $$

has kernel $\{\pm 1\}$.

Proof. The maps land in $V$. Conjugation by $u$ is an algebra automorphism, and it commutes with the conjugation, since $\overline{uvu^{-1}} = u^{\natural}\,v^{\natural}\,(u^{\natural})^{-1}$ and $\overline{u^{-1}} = (u^{\natural})^{-1}$. The vector subspace is the $-1$ eigenspace of the conjugation by (Split-Quaternion Algebra, §The Two Eigenspaces), so it is mapped to itself. The form is preserved. $N(uvu^{-1}) = N(u)N(v)N(u)^{-1} = N(v)$ by multiplicativity, since $N$ is real and nonzero on a unit. Hence $\Theta(u) \in O(V,N)$. Kernel. If $\Theta(u) = \mathrm{id}$ then $u$ commutes with every element of $V$; since $1$ and $V$ generate the algebra, $u$ is central, so $u \in \mathbb{R}^{\times}$. Conversely every nonzero scalar is in the kernel. Determinant. The map $\det\circ\,\Theta : \mathbb{H}_{\mathrm{s}}^{\times} \to \{\pm 1\}$ is a continuous homomorphism. Its domain has the two components $\{N>0\}$ and $\{N<0\}$; the first is connected, so $\det\Theta \equiv 1$ there. On the second component, the computation of (Split-Quaternion Algebra, §The Multiplication Table) gives $e_2e_1e_2 = -e_1$, $e_2e_2e_2 = e_2$, $e_2e_3e_2 = -e_3$, so $\Theta(e_2) = \operatorname{diag}(-1,1,-1)$ in the basis $e_1,e_2,e_3$ and $\det\Theta(e_2) = +1$. Hence $\det\Theta \equiv 1$ and the image lies in $SO(2,1)$. The restricted kernel. The kernel on $U$ is the centre intersected with $\{N=1\}$, that is $\{\pm 1\}$.

Corollary (The Image of the Unit Group). The image of the full unit group is

$$ \Theta(\mathbb{H}_{\mathrm{s}}^{\times}) = SO(2,1) \cong PGL_2(\mathbb{R}), $$

and the image of the norm-one group is $\Theta(U) = SO^{+}(2,1)$, the identity component.

Proof. The image of $\mathbb{H}_{\mathrm{s}}^{\times}$ is a subgroup of $SO(2,1)$ containing the image of the connected group $U$, which is connected and contains the identity, so the image contains the identity component; the domain has exactly two components and the image lies in $SO(2,1)$, which has two components, so the image is all of $SO(2,1)$. The second statement is proved below, where $\Theta(U)$ is identified with $\mathrm{SO}^{+}(2,1)$ through the double cover.

The Multiplication Operators

The adjoint action is not the only way an element acts on the algebra: left and right multiplication also realise it as operators on itself.

Definition. For $\tilde q \in \mathbb{H}_{\mathrm{s}}$ the left multiplication operator and the right multiplication operator are

$$ L_{\tilde q} : \tilde p \mapsto \tilde q \tilde p, \qquad R_{\tilde q} : \tilde p \mapsto \tilde p\tilde q . $$

Both are $\mathbb{R}$-linear endomorphisms of the four-dimensional space, and the assignments $\tilde q \mapsto L_{\tilde q}$, $\tilde q \mapsto R_{\tilde q}$ are $\mathbb{R}$-algebra (anti-)homomorphisms into $\operatorname{End}_{\mathbb{R}}(\mathbb{H}_{\mathrm{s}})$; the first is the left regular representation. Multiplication differs from the adjoint action in kind: it is not an algebra automorphism, since it does not carry $1$ to $1$, whereas the adjoint action fixes the scalar line pointwise.

Proposition. Both $L_{\tilde q}$ and $R_{\tilde q}$ have trace $4\operatorname{Sc}(\tilde q) = 4q_0$ and determinant $N(\tilde q)^2$, and $L_{\tilde q}$ is invertible if and only if $\tilde q$ is a unit.

Proof. Under the identification of the algebra with $M_2(\mathbb{R})$ — a statement about the algebra, not about a representation — left multiplication by $\tilde q$ on the four-dimensional space becomes left multiplication by the matrix $\tilde q$ on $M_2(\mathbb{R})\cong\mathbb{R}^4$, whose eigenvalues are the two eigenvalues of $\tilde q$, each with multiplicity two. Those two eigenvalues are the roots of the characteristic polynomial $\lambda^2-2\operatorname{Sc}(\tilde q)\lambda+N(\tilde q)$ (Split-Quaternion Spectral Theory), whose sum is $2q_0$ and whose product is $N(\tilde q)$. Hence $\operatorname{tr}L_{\tilde q} = 2\cdot 2q_0 = 4q_0$ and $\det L_{\tilde q} = N(\tilde q)^2$. Right multiplication $R_{\tilde q}$ is the transpose of $L_{\tilde q}$ with respect to the non-degenerate trace form $(\tilde q,\tilde p)\mapsto\operatorname{tr}(\tilde q\tilde p)$ on $M_2(\mathbb{R})$, hence has the same eigenvalues, trace and determinant. For invertibility, $L_{\tilde q}(1) = \tilde q$ shows that $L_{\tilde q} = 0$ only for $\tilde q = 0$; if $N(\tilde q)\neq0$ then $\tilde q$ is a unit and $L_{\tilde q}$ has inverse $L_{\tilde q^{-1}}$, while if $N(\tilde q) = 0$ and $\tilde q\neq0$ then $\tilde q$ is a zero divisor and there is $\tilde p\neq0$ with $\tilde q\tilde p = 0$ (Split-Quaternion Zero Divisors), so $L_{\tilde q}$ is not invertible.

Remark. The doubled eigenvalues and the square $N(\tilde q)^2$ are the operator form of the double cover: $\tilde q$ and $-\tilde q$ induce the same adjoint action on $V$ and their multiplication operators have the same determinant.

The Double Cover of $\mathrm{SO}^{+}(2,1)$

Theorem (The Double Cover). The adjoint action restricted to the norm-one group is a surjective group homomorphism

$$ \Theta : \mathrm{SL}_2(\mathbb{R}) \longrightarrow \mathrm{SO}^{+}(2,1) $$

with kernel $\{\pm 1\}$. It is therefore a double cover, and it induces an isomorphism

$$ \mathrm{SL}_2(\mathbb{R})/\{\pm 1\} = \mathrm{PSL}_2(\mathbb{R}) \cong \mathrm{SO}^{+}(2,1). $$

Proof. By the preceding theorem the homomorphism is well defined and has kernel $\{\pm 1\}$. Its differential at the identity is the representation of the Lie algebra $\mathrm{SL}_2(\mathbb{R}) = V$ with the commutator bracket (by Split-Quaternion Algebra, §The Lie Algebra Structure) on the Lie algebra $\mathrm{SO}(2,1)$ of the isometry algebra of the signature-$(2,1)$ form; it is injective because the kernel of $\Theta$ is discrete, and both Lie algebras have dimension $3$, so the differential is an isomorphism, by The Orthogonal Lie Algebra, §The Lie Algebra of Skew Transformations. The image of a Lie group homomorphism with injective differential is an open Lie subgroup of the target; since the target $\mathrm{SO}^{+}(2,1)$ is connected, an open subgroup containing the identity is the whole group, by The Lie Correspondence and the Adjoint Representation. Hence $\Theta$ is surjective, and the first isomorphism theorem for groups gives $\mathrm{SL}_2(\mathbb{R})/\{\pm 1\} \cong \mathrm{SO}^{+}(2,1)$.

Corollary (A Double Cover That Is Not the Universal Cover). The group $\mathrm{SL}_2(\mathbb{R})$ is connected but not simply connected: its fundamental group is infinite cyclic. The double cover of the theorem is therefore not the universal cover of $\mathrm{SO}^{+}(2,1)$; the universal cover is the infinite cyclic cover of $\mathrm{SL}_2(\mathbb{R})$, and the intermediate cover of degree two is the object of the theorem.

Proof. The fundamental group of $\mathrm{SL}_2(\mathbb{R})$ is computed in Matrix Groups and Classical Groups, where the maximal compact subgroup $SO(2)$ is seen to generate the fundamental group; the covering theory gives the rest.

This is a structural difference from the quaternion case and it is worth naming: the quaternion unit sphere is the three-sphere, which is simply connected, so there the double cover of $SO(3)$ is the universal cover, whereas here the double cover of $\mathrm{SO}^{+}(2,1)$ sits under an infinite tower.

Corollary (Fixed Directions). A non-identity elliptic isometry $\Theta(g(\theta))$ fixes the positive-norm direction $\mathbb{R}\xi$ and no other direction of $V$; a non-identity hyperbolic isometry $\Theta(h(t))$ fixes the negative-norm direction $\mathbb{R}\eta$ and the two isotropic directions of the plane $\eta^{\perp}$. Neither fixes any other direction.

Proof. The fixed directions of the first two kinds are computed in the two theorems below. An isometry of $V$ with three fixed directions in general position is the identity, because the form is nondegenerate and such directions span $V$; a nontrivial elliptic isometry acts on the definite plane $\xi^{\perp}$ as a rotation by the nonzero angle $2\theta$, so it fixes no direction there, and a nontrivial hyperbolic isometry acts on the indefinite plane $\eta^{\perp}$ as a hyperbolic rotation, which is the identity only for $t = 0$.

The Lorentz Group of Signature $(2,1)$

Definition. The Lorentz group of signature $(2,1)$ is the isometry group $O(V,N) \cong O(2,1)$ of the form $N = q_1^2 - q_2^2 - q_3^2$ on the three-dimensional vector subspace; its identity component is $\mathrm{SO}^{+}(2,1)$, the group of isometries of determinant $+1$ preserving the time orientation.

Theorem (The Structure of the Group). The group $O(2,1)$ has four connected components, distinguished by the signs of $\det$ and of the coordinate $q_1$ of a timelike vector; the subgroup $SO(2,1)$ of determinant $+1$ has two components, and its identity component is $\mathrm{SO}^{+}(2,1)$, a three-dimensional group isomorphic to $\mathrm{PSL}_2(\mathbb{R})$. The full group of units of the algebra maps onto $SO(2,1)$ with kernel $\mathbb{R}^{\times}$, and the norm-one group maps onto $\mathrm{SO}^{+}(2,1)$ with kernel $\{\pm 1\}$.

Proof. The components of $O(2,1)$ are described in Isometries and Orthogonal Transformations; the isomorphism $\mathrm{PSL}_2(\mathbb{R}) \cong \mathrm{SO}^{+}(2,1)$ is the double cover of the preceding section, and the statement about the images and kernels is the theorem on the adjoint representation and its corollary.

Remark (Signature $(2,1)$, not $(3,1)$). The form restricted to the vector subspace has signature $(2,1)$ by (Split-Quaternion Norm and Invertibility, §The Split-Quaternion Norm), and the group acting on the vector subspace is therefore the three-dimensional Lorentz group $\mathrm{SO}^{+}(2,1)$. The group $\mathrm{SO}^{+}(3,1)$ is the Lorentz group of four-dimensional Minkowski space and does not occur here; the split-quaternion system carries three-dimensional Lorentzian geometry and the hyperbolic plane, not four-dimensional geometry and hyperbolic three-space.

Elliptic and Hyperbolic One-Parameter Subgroups

The solutions of $\xi^2 = -1$ and of $\eta^2 = +1$ generate the two kinds of one-parameter subgroup of $U$.

Theorem (Elliptic Subgroups). Let $\xi \in V$ with $\xi^2 = -1$. Then the elements

$$ g(\theta) = \cos\theta + \xi \sin\theta, \qquad \theta \in \mathbb{R}, $$

form a subgroup of $U$ isomorphic to $SO(2)$, and

$$ \Theta(g(\theta)) \xi = \xi, \qquad \Theta(g(\theta)) : \xi^{\perp} \to \xi^{\perp} \text{ is a rotation by } 2\theta . $$

The subgroups obtained from the solutions of $\xi^2 = -1$ are the compact subgroups of $U$, and they exhaust the conjugacy classes of maximal compact subgroups.

Proof. The split-quaternion norm is $N(g(\theta)) = \cos^2\theta \cdot 1 + \sin^2\theta \cdot N(\xi) = \cos^2\theta + \sin^2\theta = 1$, using $\xi^2 = -N(\xi) = -1$; the addition formula for $g$ follows from $\xi^2 = -1$. The element $g$ commutes with $\xi$, so $\xi$ is fixed; and $\xi^{\perp}$ is the orthogonal plane of dimension two, which is definite because $\xi$ has positive norm, so that conjugation by $g$ acts on it as a rotation; the explicit computation with $\xi = e_1$ in the basis $e_2,e_3$ gives $\Theta(g(\theta))e_2 = \cos 2\theta\, e_2 + \sin 2\theta\, e_3$ and $\Theta(g(\theta))e_3 = -\sin 2\theta\, e_2 + \cos 2\theta\, e_3$, a rotation by $2\theta$, as in the quaternion case of Quaternion Rotations and Reflections, §The Homomorphism to SO(3).

Theorem (Hyperbolic Subgroups). Let $\eta \in V$ with $\eta^2 = +1$, that is $N(\eta) = -1$. Then the elements

$$ h(t) = \cosh t + \eta \sinh t, \qquad t \in \mathbb{R}, $$

form a subgroup of $U$ isomorphic to $\mathbb{R}$, and

$$ \Theta(h(t)) \eta = \eta, \qquad \Theta(h(t)) : \eta^{\perp} \to \eta^{\perp} $$

acts on the plane $\eta^{\perp}$ of signature $(1,1)$ as a hyperbolic rotation with parameter $2t$: it fixes the two isotropic lines of that plane setwise and translates along the hyperbolas $N = \text{constant}$.

Proof. The split-quaternion norm is $N(h(t)) = \cosh^2 t + \sinh^2 t \cdot 1 = \cosh^2 t - \sinh^2 t = 1$, using $N(\eta) = -1$; the addition formula follows from $\eta^2 = 1$. The element $h(t)$ commutes with $\eta$, so $\eta$ is fixed; the orthogonal plane $\eta^{\perp}$ has signature $(1,1)$ because $\eta$ is spacelike, and an isometry of a $(1,1)$ plane with a fixed nonzero vector acts as a hyperbolic rotation, fixing the two isotropic directions setwise.

Remark (versors and hyperbolic versors). The two families of the theorems are the two kinds of versor of the split-quaternion units: $\cos\theta + \xi\sin\theta$ from a root of $-1$ is a versor, and $\cosh t + \eta\sinh t$ from a unit of square $+1$ is a hyperbolic versor, an element whose earliest instances are Cockle's tessarines. On the plane $\eta^{\perp}$ of Lorentzian signature the second family is the one-parameter group of a boost of rapidity $2t$ at the generator $\eta$, so the hyperbolic versor is the exponential form of the boost; the closed form and the naming are in Split-Quaternion Exponential and Lie Group Structure.

Corollary (The Trichotomy of Subgroups). The subgroup generated by a solution of $\xi^2 = -1$ is compact and consists of elements without real eigenvalues; the subgroup generated by a solution of $\eta^2 = +1$ is non-compact and its non-identity elements have the two real eigenvalues $e^{\pm t}$ of the isometry. The three families correspond to the three conjugacy classes of one-parameter subgroups of $U$: the elliptic class, the hyperbolic class, and the parabolic class, the last consisting of the subgroups generated by the nilpotents of $V$ and acting by parabolic isometries fixing a single isotropic line.

Proof. The first two statements are the computations of the two theorems; the parabolic case uses the nilpotents of (Split-Quaternion Zero Divisors, §Nonzero Nilpotents), whose exponentials are the elements $1 + t\xi$ with $\xi^2 = 0$ and $N(\xi) = 0$.

The Trichotomy of Timelike, Lightlike and Spacelike Elements

The form $N$ is isotropic, so the trichotomy is real: all three classes are nonempty, unlike in the quaternion case, where the form is definite and only the analogue of the timelike class occurs.

Definition. A nonzero element $v \in V$ is timelike when $N(v) > 0$, lightlike when $N(v) = 0$, and spacelike when $N(v) < 0$. The lightlike elements are the nilpotents of (Split-Quaternion Zero Divisors, §Nonzero Nilpotents).

Theorem (The Orbits). The action of $\mathrm{SO}^{+}(2,1)$ on $V$ has the following orbits.

Orbit	Criterion	Structure
$\{0\}$	$v = 0$	one point
two timelike orbits	$N(v) = 1$, the two sheets $q_1 \geq 1$ and $q_1 \leq -1$	each a copy of the hyperbolic plane
two lightlike orbits	$v \neq 0$, $N(v) = 0$	the two nappes of the cone, each a homogeneous space of dimension $2$
one spacelike orbit	$N(v) = -1$	the one-sheeted hyperboloid, a homogeneous space of dimension $2$

Each nontrivial orbit is a level set of $N$ scaled to $\pm 1$ or $0$, and the stabiliser of a timelike point is a copy of $SO(2)$, the stabiliser of a spacelike point is a copy of $SO(1,1) \cong \mathbb{R}$, and the stabiliser of a lightlike point is the one-parameter unipotent group of null transvections, isomorphic to $\mathbb{R}$.

Proof. The form and its action are those of the preceding sections; the level sets are invariant because $\Theta$ preserves $N$, and transitivity on each level set is the standard transitivity of the Lorentz group on each hyperboloid, proved in Isometries and Orthogonal Transformations, together with the orbit–stabiliser theorem. The stabiliser of a timelike vector is the group of isometries of the positive-definite orthogonal complement, namely $SO(2)$; that of a spacelike vector is the group of isometries of the signature-$(1,1)$ complement, namely $SO(1,1)$; and that of a lightlike vector preserves the radical $\mathbb{R}v$ of the orthogonal complement and acts trivially on the one-dimensional quotient $v^{\perp}/\mathbb{R}v$, so that its Lie algebra is the line $\{X \in \mathrm{SO}(2,1) : Xv = 0\}$, of dimension one: every orbit in the table is two-dimensional and the group is three-dimensional, so every stabiliser is one-dimensional, the three of them being $SO(2)$, $SO(1,1)$ and the unipotent group. (The stabiliser of a lightlike direction, which is a point of the boundary of the hyperbolic plane rather than a vector of the cone, is the larger two-dimensional solvable group, the affine group of the line.)

Corollary (The Two Sheets and the Hyperbolic Plane). Each sheet of $N = 1$ is a copy of the hyperbolic plane, and the identity component of its isometry group is $\mathrm{SO}^{+}(2,1) \cong \mathrm{PSL}_2(\mathbb{R})$. The model is built on the split-quaternions in Split-Quaternions and Hyperbolic Geometry, which carries the geometry; the present article supplies the group action, not the model.

Proof. The orbit of a timelike unit vector is $\mathrm{SO}^{+}(2,1)/SO(2)$, and this quotient is a model of the hyperbolic plane by Hyperbolic Geometry; the identification of the sheets with the hyperboloid model is (Split-Quaternion Norm and Invertibility, §Isotropy) combined with the transitivity above.

Corollary (The Sign of the Split-Quaternion Norm and the Fixed Vectors). An isometry $\Theta(g)$ with a nonzero fixed vector of $V$ has that vector timelike, lightlike or spacelike according to its type: elliptic isometries fix a timelike vector if they are non-trivial, hyperbolic isometries fix a spacelike vector and the two lightlike directions of its orthogonal plane, and parabolic isometries fix a unique lightlike line and no other direction.

Proof. The fixed vectors of $g$ are the eigenvectors of $g$ in $V$, and the stated eigenvectors are those computed in the three cases of the one-parameter subgroups.

Invariants and the Action on the Distinguished Subspaces

Conjugation by a unit is an automorphism of the whole algebra, not only a linear map of $V$ (Split-Quaternion Automorphisms and Derivations), so it preserves the two summands of $\mathbb{H}_{\mathrm{s}} = S\oplus V$ separately and never mixes them: it acts trivially on the scalar line $S$, which it fixes pointwise, and by the Lorentz action on $V$. On the whole algebra the invariants of the adjoint action are therefore the scalar part $q_0$ and the split-quaternion norm $N(\tilde q)$, the first because $S$ is fixed and the second because the action is an automorphism; this is why the whole group-theoretic content of the operator sits on the vector subspace.

The adjoint action on the distinguished subspaces is as follows.

subspace	image under $\Theta(g)$	preserved?	remark
$S = \mathbb{R}\cdot 1$	$S$	yes, pointwise	the centre is fixed
$V = \operatorname{span}\{e_1,e_2,e_3\}$	$V$	yes	the Lorentz action of signature $(2,1)$
a line $\mathbb{R} v \subset V$	a line $\mathbb{R}\,\Theta(g)v$	yes as a class	the sign of $N(v)$ is preserved
$\mathbb{D}_2 = \operatorname{span}\{1,e_2\}$	a conjugate split-complex plane	only if $g$ normalises it	automorphisms permute the split-complex planes
$\mathbb{D}_3 = \operatorname{span}\{1,e_3\}$	a conjugate split-complex plane	only if $g$ normalises it	as above
$\mathbb{H}_{\mathrm{s}} \tilde\pi_\pm$	a minimal left ideal	only if $g$ normalises it	idempotents map to idempotents

The first three rows are the content of the preceding sections. The remaining rows record that an automorphism sends a subalgebra to a subalgebra of the same isomorphism type: the split-complex planes are permuted among their conjugates, since the split Cartan subalgebras of $M_2(\mathbb{R})$ are conjugate, and a unit normalises a given plane exactly when it preserves the pair of isotropic lines of that plane. The idempotents form a single orbit-like set under the automorphism group and the minimal left ideals are permuted accordingly; this is developed alongside the idempotent theory in Split-Quaternion Ideals and Peirce Decomposition.

The Relation to the Polar Representation

Every nonzero split-quaternion has a polar representation $\tilde q = \rho u$ with $\rho = \sqrt{|N(\tilde q)|} > 0$ and $u$ a unit of norm $\pm1$ (Split-Quaternion Polar Element Representation). In it the operator content of $\tilde q$ splits: the positive scale contributes the scalar operator $L_\rho = \rho\,\mathrm{id}$, and the unit contributes the adjoint automorphism $\Theta(u)$ on $V$, which is the Lorentz transformation the element carries. Left multiplication by $\tilde q$ is therefore the composite of a scaling and an operator whose restriction to $V$ is that Lorentz transformation: $L_{\tilde q} = \rho\,L_u$, and the adjoint action sees only $u$. The explicit polar factor in each of the three sign cases $N>0$, $N<0$, $N=0$ is the subject of Split-Quaternion Polar Element Representation.

Comparison with the Quaternion and Split-Biquaternion Cases

The Quaternion Case

	$\mathbb{H}$	$\mathbb{H}_{\mathrm{s}}$
unit group	$Sp(1) \cong S^3$, compact, simply connected	$\mathrm{SL}_2(\mathbb{R})$, non-compact, not simply connected
norm levels	$N = 1$ only	$N = \pm 1$, two components
isometry group produced	$SO(3)$, by the adjoint action on $\operatorname{Im}\mathbb{H}$	$\mathrm{SO}^{+}(2,1)$, by the adjoint action on $V$
cover	universal double cover	double cover, not universal
form on the vector part	definite, signature $(3,0)$	isotropic, signature $(2,1)$
trichotomy	only one class, all nonzero vectors equivalent	timelike, lightlike, spacelike, all nonempty

The quaternion column is the classical description of the unit quaternions as the double cover of the rotation group, recorded in Quaternion Rotations and Reflections, §The Double Cover of SO(3). The single structural cause of every difference is the sign pattern: a definite form gives a compact sphere and one class of vectors, an isotropic form gives a non-compact hyperboloid and the full trichotomy.

The Split-Biquaternion Case

The eight-dimensional algebra $\mathbb{H}_{\mathbb{D}}$ of the notation table is a later system of Part V, treated under Split-Biquaternions, and nothing of it is used here. The one thing worth stating from the conventions is a warning about size: the present article's isometry group is the three-dimensional $\mathrm{SO}^{+}(2,1)$ acting on the three-dimensional vector subspace of a four-dimensional algebra, and the eight-dimensional relative is a different system with its own, larger, geometry, treated later. The name split quaternions belongs to the four-dimensional algebra of this category and not to $\mathbb{H}_{\mathbb{D}}$.

Summary

The norm-one group of the split-quaternion algebra is $U \cong \mathrm{SL}_2(\mathbb{R})$, connected; the group of units of norm $\pm 1$ has the two components $U$ and the split-quaternion norm $-1$ component interchanged by $e_2$; and the full group of units $\{N \neq 0\}$ has centre $\mathbb{R}^{\times}$.

Conjugation by a unit preserves the vector subspace and the form, giving a homomorphism from the group of units onto $\mathrm{SO}(2,1) \cong \mathrm{PGL}_2(\mathbb{R})$ with kernel $\mathbb{R}^{\times}$ — onto the full group, not only its identity component, because a negative-norm unit such as $e_2$ acts on $V$ as $\operatorname{diag}(-1,1,-1)$, which reverses the sheets — and a homomorphism $\mathrm{SL}_2(\mathbb{R}) \to \mathrm{SO}^{+}(2,1)$ with kernel $\{\pm 1\}$, hence a double cover and an isomorphism $\mathrm{PSL}_2(\mathbb{R}) \cong \mathrm{SO}^{+}(2,1)$. The double cover is not the universal cover, because $\mathrm{SL}_2(\mathbb{R})$ has infinite cyclic fundamental group. The Lorentz group of the system is the three-dimensional $\mathrm{SO}(2,1)$ of the signature-$(2,1)$ form, with identity component $\mathrm{SO}^{+}(2,1)$, and not $\mathrm{SO}(3,1)$.

Besides conjugation, an element acts by left multiplication $L_{\tilde q}(\tilde p) = \tilde q\tilde p$ and right multiplication $R_{\tilde q}(\tilde p) = \tilde p\tilde q$, operators of trace $4q_0$ and determinant $N(\tilde q)^2$, with $L_{\tilde q}$ invertible exactly for a unit. On the whole algebra the adjoint action preserves the scalar part and the norm and never mixes $S$ with $V$, so its invariants are $q_0$ and $N(\tilde q)$; it fixes $S$ pointwise, preserves $V$, and permutes the split-complex planes and the minimal left ideals among their conjugates. In the polar form $\tilde q = \rho u$ the scale contributes the scalar operator $\rho\,\mathrm{id}$ and the unit the Lorentz transformation $\Theta(u)$, so that $L_{\tilde q} = \rho\,L_u$ and conjugation sees only $u$.

The roots of $\xi^2 = -1$ generate the compact (elliptic) one-parameter subgroups, isomorphic to $SO(2)$ and fixing a timelike direction and rotating its orthogonal plane; the roots of $\eta^2 = +1$ generate the non-compact (hyperbolic) subgroups, fixing a spacelike direction and acting as hyperbolic rotations on its orthogonal plane of signature $(1,1)$; the nilpotents generate the parabolic subgroups. Under the norm-one group $U \cong \mathrm{SL}_2(\mathbb{R})$, the nonzero vectors of the vector subspace fall into two timelike orbits (the sheets of the hyperboloid $N=1$, each a hyperbolic plane), two lightlike orbits (the nappes of the cone) and one spacelike orbit; the four sheets and nappes merge in pairs under the negative-norm units, so that the full group of units has one orbit of each sign type. The trichotomy is nonempty because the form is isotropic. The quaternion case has a compact simply connected unit sphere, a universal double cover of $SO(3)$, a definite vector form and a single class of vectors; the eight-dimensional relative is a later system of Part V, named here only.

Summary of Notation

Symbol	Meaning	Article
$U = \{N=1\}$	the norm-one group, $\cong \mathrm{SL}_2(\mathbb{R})$	Split-Quaternion Norm and Invertibility
$U^{\pm} = \{N = \pm 1\}$	$\mathrm{SL}_2^{\pm}(\mathbb{R})$, two components	this article
$\mathbb{H}_{\mathrm{s}}^{\times} = \{N \neq 0\}$	the group of units	Split-Quaternion Norm and Invertibility
$\Theta(u)v = uvu^{-1}$	the adjoint action on $V$	this article
$L_{\tilde q}(\tilde p) = \tilde q\tilde p$, $R_{\tilde q}(\tilde p) = \tilde p\tilde q$	left and right multiplication operators; trace $4q_0$, determinant $N(\tilde q)^2$	this article
$(V,N) \cong \mathbb{R}^{2,1}$	the vector subspace with its signature-$(2,1)$ form	Split-Quaternion Algebra
$O(2,1)$, $SO(2,1)$, $\mathrm{SO}^{+}(2,1)$	the Lorentz group, its determinant-one part, its identity component	this article
$\mathrm{SL}_2(\mathbb{R}) \to \mathrm{SO}^{+}(2,1)$	the double cover	this article
$\mathrm{PSL}_2(\mathbb{R}) \cong \mathrm{SO}^{+}(2,1)$	the isomorphism induced by the cover	this article
$g(\theta) = \cos\theta + \xi\sin\theta$	the elliptic one-parameter subgroup	this article
$h(t) = \cosh t + \eta\sinh t$	the hyperbolic one-parameter subgroup	this article
timelike, lightlike, spacelike	$N>0$, $N=0$, $N<0$ on $V$	this article
nilpotents of $V$	the lightlike elements and the parabolic generators	Split-Quaternion Zero Divisors