The Pin and Spin Groups of the General Quaternionic Algebra

Introduction

The twisted two-sided operator $\Theta_{\tilde Q_5}(\tilde Q_1)=\tilde Q_5\tilde Q_1\tilde Q_5^{\natural}$ is an automorphism of the general quaternionic bilinear form exactly when $\langle\tilde Q_5\rangle_{\natural}=\pm1$, and it is multiplicative in the parameter, $\Theta_{\tilde Q_5}\Theta_{\tilde Q_2}=\Theta_{\tilde Q_5\tilde Q_2}$. The set of parameters with $N=\pm1$ is therefore a group that acts on the algebra by automorphisms through $\Theta$, and the assignment is two-to-one; this article reads the group, its two constituents, the versors and the sandwich they carry on the vector subspace, and the reflections of that subspace.

The two constituents are the pin group $\mathrm{Pin}=\{\tilde Q_5\in\mathbb{B}^\times:\langle\tilde Q_5\rangle_{\natural}=\pm1\}$ and the spin group $\mathrm{Spin}=\{\tilde Q_5\in\mathbb{B}^\times:\langle\tilde Q_5\rangle_{\natural}=1\}$, the latter of index two. The twisted operator is Two-Sided Operators on the General Quaternionic Algebra; the one-sided factors are One-Sided Operators on the General Quaternionic Algebra; the form and its orthogonal group are The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra; the unit group into which both groups sit is The Biquaternion Unit Group as a Topological Group; the general Clifford theory of the pin and spin groups is The Clifford, Pin and Spin Groups with Signed Inner Conjugation, of which this is the biquaternion instance; and the vector subspace and the reflection are Introduction to the Remarkable Subspaces and Biquaternion Rotations and Lorentz Transformations.

The Two Groups

Definition. The pin group and the spin group of the general quaternionic bilinear form are

$$ \mathrm{Pin}=\{\tilde Q_5\in\mathbb{B}^\times:\langle\tilde Q_5\rangle_{\natural}=\pm1\},\qquad \mathrm{Spin}=\{\tilde Q_5\in\mathbb{B}^\times:\langle\tilde Q_5\rangle_{\natural}=1\}=\ker N . $$

Proposition (they are groups, and $\mathrm{Spin}$ has index two). Both are subgroups of the unit group, closed under multiplication and inverse; $\mathrm{Spin}=\mathrm{Pin}\cap\ker N$ is normal of index two in $\mathrm{Pin}$, and the quotient is $\mathrm{Pin}/\mathrm{Spin}\cong\{\pm1\}$ through the norm.

Proof. The norm is multiplicative and central, so if $\langle\tilde Q_5\rangle_{\natural},\langle\tilde Q_2\rangle_{\natural}\in\{\pm1\}$ then $\langle\tilde Q_5\tilde Q_2\rangle_{\natural}\in\{\pm1\}$ and $\langle\tilde Q_5^{-1}\rangle_{\natural}=\langle\tilde Q_5\rangle_{\natural}^{-1}=\langle\tilde Q_5\rangle_{\natural}$; this makes $\mathrm{Pin}$ a group and $\mathrm{Spin}$, its norm-one part, a subgroup. The norm is a homomorphism onto $\{\pm1\}$ on $\mathrm{Pin}$, hence $\mathrm{Pin}/\mathrm{Spin}\cong\{\pm1\}$.

Remark (the two constituents and the real shadow). $\mathrm{Pin}$ is the disjoint union $\mathrm{Spin}\sqcup\{N=-1\}$: the norm-one shell, a complex hypersurface of complex dimension $3$, and the norm-minus-one shell, its $i$-translate, since $\langle i\tilde Q_5\rangle_{\natural}=i^{2}\langle\tilde Q_5\rangle_{\natural}=-\langle\tilde Q_5\rangle_{\natural}$ and the two shells are the same set moved by the central scalar $i$. On the real quaternion subspace $\mathbb{H}_{\mathbb{B}}$ the norm is positive definite, $\langle\tilde Q\rangle_{\natural}=\sum_\mu q_\mu^{2}\geq0$, so only the condition $\langle\cdot\rangle_{\natural}=1$ is met there, on the norm-one set $S^{3}\cong SU(2)\cong\mathrm{Spin}(3)$, and no element of $\mathbb{H}_{\mathbb{B}}$ has norm minus one; the norm-minus-one elements need the imaginary directions, the simplest being $ie_t$. On the complex algebra both shells meet the null cone.

Proposition (the matrix identifications). Under the isomorphism $\mathsf{M}_2:\mathbb{B}\to M_2(\mathbb{C})$ of Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions, with $\det\mathsf{M}_2(\tilde Q_5)=\langle\tilde Q_5\rangle_{\natural}$,

$$ \boxed{\; \mathrm{Spin}\cong SL_2(\mathbb{C}),\qquad \mathrm{Pin}\cong\{\tilde Q_5\in GL_2(\mathbb{C}):\det\mathsf{M}_2(\tilde Q_5)^{2}=1\} \;}, $$

and $\mathrm{Spin}$ is the norm-one subgroup of real dimension $6$ of The Biquaternion Unit Group as a Topological Group.

Proof. The determinant identity is the norm, so $\langle\cdot\rangle_{\natural}=1$ is $\det\mathsf{M}_2=1$; the pin condition is $\det\mathsf{M}_2=\pm1$, that is $\det^{2}=1$. The dimension and connectedness are those of the norm-one group.

The Action and the Two-to-One Cover

Theorem (the cover). The assignment $\tilde Q_5\mapsto\Theta_{\tilde Q_5}$ is a group homomorphism from $\mathrm{Pin}$ to the group of automorphisms of the general quaternionic bilinear form, with kernel $\{\pm e_t\}$; it is therefore a two-to-one cover of its image,

$$ \boxed{\; \mathrm{Pin}/\{\pm e_t\}\ \cong\ \{\Theta_{\tilde Q_5}:\langle\tilde Q_5\rangle_{\natural}=\pm1\}\ \subset\ O(\mathbb{B},N) \;}. $$

Proof. The composition law $\Theta_{\tilde Q_5}\Theta_{\tilde Q_2}=\Theta_{\tilde Q_5\tilde Q_2}$ makes the assignment a homomorphism; the preservation condition is $\langle\Theta_{\tilde Q_5}\tilde Q_1\rangle_{\natural}=\langle\tilde Q_5\rangle_{\natural}^{2}\langle\tilde Q_1\rangle_{\natural}$, which is preservation exactly for $\langle\tilde Q_5\rangle_{\natural}=\pm1$; and the kernel is $\{\pm e_t\}$ by the injectivity up to sign of the assignment, $\Theta_{\tilde Q_5}=\Theta_{\tilde Q_2}$ for units being $\tilde Q_2=\pm\tilde Q_5$.

Proposition (the norm-one image is the rotation group of the vector subspace). On $\mathrm{Spin}$ the twisted operator is the inner automorphism, $\Theta_{\tilde Q_5}=\mathrm{Ad}_{\tilde Q_5}$ for $\langle\tilde Q_5\rangle_{\natural}=1$, and the image is the group of inner automorphisms of the algebra,

$$ \boxed{\; \Theta(\mathrm{Spin})\ \cong\ \mathrm{Spin}/\{\pm e_t\}\ \cong\ \mathrm{Inn}(\mathbb{B})\ \cong\ PGL_2(\mathbb{C})\ \cong\ SO_3(\mathbb{C}) \;}, $$

of complex dimension $3$, acting on $\mathbb{B}=\mathbb{C}_{\mathbb{B}}\oplus V$ by fixing the centre pointwise and rotating the vector subspace. Under this action $\mathrm{Spin}\cong SL_2(\mathbb{C})$ is the classical two-to-one cover of the rotation group.

Proof. On the norm-one shell $\tilde Q_5^{\natural}=\tilde Q_5^{-1}$ and the operator is $\langle\tilde Q_5\rangle_{\natural}\mathrm{Ad}_{\tilde Q_5}=\mathrm{Ad}_{\tilde Q_5}$, an algebra automorphism; a conjugation fixes the centre and preserves the trace-zero subspace $V$, and every automorphism of $\mathbb{B}\cong M_2(\mathbb{C})$ is inner by the Skolem–Noether theorem, while a scalar may be rescaled to norm one over $\mathbb{C}$, so the image is the whole inner automorphism group; $\mathrm{Inn}(M_2(\mathbb{C}))=PGL_2(\mathbb{C})$, and $PGL_2(\mathbb{C})\cong SO_3(\mathbb{C})$ is the classical isomorphism, acting on the three-dimensional traceless subspace by the rotations preserving the norm. The kernel is again $\{\pm e_t\}$, which is the classical two-to-one cover.

Remark (what the norm-minus-one part adds). The elements of norm minus one give operators that are multiplicative up to the sign, $\Theta_{\tilde Q_5}(\tilde Q_1)\Theta_{\tilde Q_5}(\tilde Q_3)=\langle\tilde Q_5\rangle_{\natural}\Theta_{\tilde Q_5}(\tilde Q_1\tilde Q_3)=-\Theta_{\tilde Q_5}(\tilde Q_1\tilde Q_3)$; they are automorphisms of the form all the same, and they are exactly the coset that makes the cover $\mathrm{Pin}\to\{\Theta\}$ larger than the rotation cover. The two cosets of $\mathrm{Spin}$ in $\mathrm{Pin}$ cover the automorphisms and the twisted maps.

Versors and the Sandwich Action

The twist $\natural$ is the sign character of the algebra: it is $+1$ on the centre $\mathbb{C}_{\mathbb{B}}$ and $-1$ on the quaternion vector subspace $V$. The elements on which the twist is $-1$ play, in this algebra, the role that the vectors of a quadratic space play in the general theory of the signed inner conjugation, and the two-sided operator of such an element is the sandwich of that theory. In the grade dictionary of the Clifford algebra the elements of $V$ are the grade-two elements and the geometric grade-one vectors are the imaginary ones $i\vec e_k$; this article keeps the quaternion naming of the group, in which $\vec e_k$ is a vector, and states the dictionary once here so that the two readings are not confused.

Definition. A versor of the algebra is a product $\tilde Q_5=\tilde{Q}_{4,1}\tilde{Q}_{4,2}\cdots\tilde{Q}_{4,k}$ of elements of $V$ with $\langle\tilde{Q}_{4,j}\rangle_{\natural}\neq0$; the Lipschitz group is the group that the non-isotropic vectors of $V$ generate.

Proposition (versors are units, and the norm multiplies). Every versor is a unit, and

$$ \langle\tilde{Q}_{4,1}\tilde{Q}_{4,2}\cdots\tilde{Q}_{4,k}\rangle_{\natural}=\langle\tilde{Q}_{4,1}\rangle_{\natural}\langle\tilde{Q}_{4,2}\rangle_{\natural}\cdots \langle\tilde{Q}_{4,k}\rangle_{\natural}; $$

after division by a square root of its norm every versor lies in $\mathrm{Pin}$, and in $\mathrm{Spin}$ exactly when its norm is one.

Proof. A vector $\tilde{Q}_4$ with $\langle\tilde{Q}_4\rangle_{\natural}\neq0$ is invertible, with $\tilde{Q}_4^{-1}=\tilde{Q}_4^{\natural}/\langle\tilde{Q}_4\rangle_{\natural}$, because $\tilde{Q}_4\tilde{Q}_4^{\natural}=\langle\tilde{Q}_4\rangle_{\natural}e_t$; a product of units is a unit. The norm is multiplicative on the whole algebra, so it multiplies over the factors, and the normalised element $\tilde Q_5/\langle\tilde Q_5\rangle_{\natural}^{1/2}$ has norm $\langle\tilde Q_5\rangle_{\natural}\big/\bigl(\langle\tilde Q_5\rangle_{\natural}^{1/2}\bigr)^{2}=1$, a square root of a nonzero complex number being available.

Proposition (the sandwich on the vector subspace). For every unit $\tilde Q_5$ the twisted operator preserves the vector subspace and acts on it by a similarity with multiplier $\langle\tilde Q_5\rangle_{\natural}^{2}$:

$$ \Theta_{\tilde Q_5}(V)\subseteq V,\qquad \langle\Theta_{\tilde Q_5}\tilde W\rangle_{\natural}=\langle\tilde Q_5\rangle_{\natural}^{2}\langle\tilde W\rangle_{\natural}\qquad(\tilde W\in V). $$

For $\tilde Q_5$ in the pin group the restriction is therefore an automorphism of the restricted form, of determinant $\langle\tilde Q_5\rangle_{\natural}^{3}$ in the basis $\vec e_x,\vec e_y,\vec e_z$.

Proof. The operator is $\Theta_{\tilde Q_5}=\langle\tilde Q_5\rangle_{\natural}\mathrm{Ad}_{\tilde Q_5}$, a central scalar times an inner automorphism. An automorphism of the algebra preserves the centre and the trace-zero subspace $V$, so $\Theta_{\tilde Q_5}(V)\subseteq V$; an inner automorphism preserves the determinant, hence the norm, and the scalar contributes its square, giving $\langle\Theta_{\tilde Q_5}\tilde W\rangle_{\natural}=\langle\tilde Q_5\rangle_{\natural}^{2}\langle\tilde W\rangle_{\natural}$. In the three-dimensional basis the determinant of a scalar times a map is the cube of the scalar, and the inner automorphism has determinant one on $V$; the determinant of the restriction is therefore $\langle\tilde Q_5\rangle_{\natural}^{3}$, of sign $\langle\tilde Q_5\rangle_{\natural}$.

Corollary (the half-turn carried by a vector). For $\tilde{Q}_4\in V$ with $\langle\tilde{Q}_4\rangle_{\natural}\neq0$ the sandwich is the half-turn about the line $\mathbb{C}\tilde{Q}_4$, composed with the homothety of ratio $\langle\tilde{Q}_4\rangle_{\natural}$: writing $\tau_{\tilde{Q}_4}$ for the half-turn,

$$ \Theta_{\tilde{Q}_4}(\tilde W)=\langle\tilde{Q}_4\rangle_{\natural}\,\tau_{\tilde{Q}_4}(\tilde W),\qquad \tau_{\tilde{Q}_4}(\tilde W)=2\,\frac{\langle\tilde W,\tilde{Q}_4\rangle_{\natural}}{\langle\tilde{Q}_4\rangle_{\natural}}\,\tilde{Q}_4-\tilde W . $$

Thus $\Theta_{\tilde{Q}_4}$ acts on $\tilde{Q}_4$ by $\langle\tilde{Q}_4\rangle_{\natural}$ and on its orthogonal complement by $-\langle\tilde{Q}_4\rangle_{\natural}$: for $\langle\tilde{Q}_4\rangle_{\natural}=1$ it is the half-turn, fixing the line and negating the complement, of determinant $+1$; for $\langle\tilde{Q}_4\rangle_{\natural}=-1$ the homothety and the half-turn combine into the honest reflection $\rho_{\tilde{Q}_4}$ itself, negating the line, fixing the complement, of determinant $-1$.

Proof. Insert the reflection formula of the next section, $\Theta_{\tilde{Q}_4}=-\langle\tilde{Q}_4\rangle_{\natural}\rho_{\tilde{Q}_4}$, and $\rho_{\tilde{Q}_4}(\tilde W)=\tilde W-2\langle\tilde W,\tilde{Q}_4\rangle_{\natural}\langle\tilde{Q}_4\rangle_{\natural}^{-1}\tilde{Q}_4$, writing $\tau_{\tilde{Q}_4}=-\rho_{\tilde{Q}_4}$ for the half-turn.

Remark (why every unit preserves the vector subspace here). In the general theory the condition $\Theta(V)\subseteq V$ defines the Clifford group, and it is not a consequence of the norm condition: over the real form of signature $(1,1)$ the unit of norm one recorded in The Clifford, Pin and Spin Groups with Signed Inner Conjugation has a signed inner conjugation that carries a vector outside the space of vectors. That warning does not bite in the biquaternion algebra. Here $\mathbb{B}\cong M_2(\mathbb{C})$ is a full matrix algebra over an algebraically closed field, every automorphism of it is inner by Skolem–Noether, and a central scalar times an inner automorphism preserves $V$; the Clifford group is therefore the whole unit group $\mathbb{B}^{\times}$, and inside it the pin group is exactly the norm shell, not a proper subgroup of it.

Proposition (the groups generated by the versors). The vectors of $V$ of norm one lie in $\mathrm{Spin}$ and act on $V$ by the half-turns about their lines; products of two of them act by the rotations, and by Cartan–Dieudonné in dimension three every rotation of $V$ is a product of two reflections and corresponds in the spin group to a product of two unit vectors (The Clifford, Pin and Spin Groups with Signed Inner Conjugation, §Cartan–Dieudonné and the Reflection Length). The non-isotropic vectors of $V$ therefore generate $\mathrm{Pin}$, their norm-one part generates $\mathrm{Spin}$, and the image of the latter is the rotation group $\Theta(\mathrm{Spin})\cong SO_3(\mathbb{C})$ of the vector subspace.

Proof. A norm-one vector $\tilde{Q}_4$ lies in $\mathrm{Pin}$, indeed in $\mathrm{Spin}$; by the corollary its image on $V$ fixes $\tilde{Q}_4$ and negates the orthogonal complement, which is the reflection $\rho_{\tilde{Q}_4}$ up to the sign, a half-turn. Two half-turns compose to a rotation, $\Theta_{\tilde{Q}_4}\Theta_{\tilde W}=\Theta_{\tilde{Q}_4\tilde W}$ with $\langle\tilde{Q}_4\tilde W\rangle_{\natural}=1$, and the quoted theorem of the general article supplies every rotation as such a product; the resulting group of operators has been identified with $\mathrm{Inn}(\mathbb{B})\cong SO_3(\mathbb{C})$ above.

The Reflection

Proposition (the reflection on the vector subspace). For $\tilde{Q}_4\in V$ with $\langle\tilde{Q}_4\rangle_{\natural}\neq0$ the twisted operator fixes the centre up to the norm and equals the reflection up to the norm and the sign:

$$ \Theta_{\tilde{Q}_4}(e_t)=\langle\tilde{Q}_4\rangle_{\natural}e_t,\qquad \Theta_{\tilde{Q}_4}(\tilde W)=-\langle\tilde{Q}_4\rangle_{\natural}\,\rho_{\tilde{Q}_4}(\tilde W)\qquad(\tilde W\in V). $$

In particular, for a vector of norm one, $\Theta_{\tilde{Q}_4}(\tilde W)=-\rho_{\tilde{Q}_4}(\tilde W)$ on $V$: the twisted operator carries the reflection composed with the sign, and on the norm-one shell it is the plain inner conjugation, so the sign cannot be removed by a change of the description.

Proof. $\Theta_{\tilde{Q}_4}(e_t)=\tilde{Q}_4\tilde{Q}_4^{\natural}=\langle\tilde{Q}_4\rangle_{\natural}e_t$. For $\tilde W\in V$, write the pure quaternions as vectors and use the classical identities $\tilde{Q}_4\tilde W=-\langle\tilde{Q}_4,\tilde W\rangle_{\natural}e_t+(\tilde{Q}_4\times\tilde W)$ and $(\tilde{Q}_4\times\tilde W)\times\tilde{Q}_4=\langle\tilde{Q}_4\rangle_{\natural}\tilde W-\langle\tilde{Q}_4,\tilde W\rangle_{\natural}\tilde{Q}_4$ of the quaternion product; then $\tilde{Q}_4\tilde W\tilde{Q}_4^{\natural}=-\tilde{Q}_4\tilde W\tilde{Q}_4=\langle\tilde{Q}_4\rangle_{\natural}\tilde W-2\langle\tilde{Q}_4,\tilde W\rangle_{\natural}\tilde{Q}_4$, which is $\langle\tilde{Q}_4\rangle_{\natural}\rho_{\tilde{Q}_4}(\tilde W)$; inserting the sign of $\tilde{Q}_4^{\natural}=-\tilde{Q}_4$ gives the displayed identity.

Remark (the minus is the general one). The appearance of $-\rho_{\tilde{Q}_4}$ rather than $\rho_{\tilde{Q}_4}$ is the phenomenon the Clifford layer records: for an element of the odd part the plain inner conjugation returns the reflection composed with $-\mathrm{id}$, and it is the twisted member of the family, with the twist in the left factor, that returns the reflection itself. On the norm-one shell the operator $\Theta$ here coincides with the plain inner conjugation, so it carries the minus; the corresponding signed left twist of the general theory differs from $\Theta$ by the norm normalisation and is discussed in The Clifford, Pin and Spin Groups with Signed Inner Conjugation. The half-turn reading of the same identity, and the generation of the rotation group by the versors, are in the section above.

Worked Examples

A spin element of the centre. For $\tilde Q_5=\pm e_t$, $N=1$ and $\Theta_{\tilde Q_5}=\mathrm{id}$; the two elements give the same operator, which is the kernel of the cover.

A spin element of norm one and not central. For $\tilde Q_5=(e_t+\vec e_x)/\sqrt2$, $\langle\cdot\rangle_{\natural}=1$ and $\Theta_{\tilde Q_5}=\mathrm{Ad}_{\tilde Q_5}$ is an inner automorphism, fixing the centre and rotating the vector subspace; its inverse in the group is $\tilde Q_5^{\natural}=\tilde Q_5^{-1}=(e_t-\vec e_x)/\sqrt2$.

A pin element of norm minus one. For $\tilde Q_5=ie_t$, $\langle\cdot\rangle_{\natural}=-1$ and $\Theta_{\tilde Q_5}=-\mathrm{id}$; the operator is a form-preserving map with $\det=+1$ and is not an automorphism: it is the twisted coset.

A vector of norm one. For $\tilde{Q}_4=\vec e_x$ the element lies in $\mathrm{Spin}$, and $\Theta_{\tilde{Q}_4}$ fixes $e_t$ and acts on $V$ by $\Theta_{\tilde{Q}_4}(\vec e_x)=\vec e_x$, $\Theta_{\tilde{Q}_4}(\vec e_y)=-\vec e_y$, $\Theta_{\tilde{Q}_4}(\vec e_z)=-\vec e_z$; the reflection $\rho_{\tilde{Q}_4}$ acts by $-\Theta_{\tilde{Q}_4}$, that is $-\vec e_x$ on $\tilde{Q}_4$ and identity on its complement, as the proposition prescribes. The half-turn reading is the same statement: $\Theta_{\tilde{Q}_4}$ is the half-turn about the line $\mathbb{C}\vec e_x$, of determinant $+1$ on $V$.

A vector of norm minus one. For $\tilde{Q}_4=i\vec e_x$, $\langle\tilde{Q}_4\rangle_{\natural}=-1$, and $\Theta_{\tilde{Q}_4}=\rho_{\tilde{Q}_4}$ is the honest reflection: it negates $i\vec e_x$, fixes the orthogonal complement, and has determinant $-1$ on $V$; the homothety of ratio $-1$ and the half-turn have combined into a reflection, as the corollary prescribes. The element lies in the odd part of the pin group and not in $\mathrm{Spin}$.

A versor of length two. For $\tilde Q_5=\vec e_x\vec e_y=\vec e_z$, the product of the two norm-one vectors $\vec e_x$ and $\vec e_y$, the norm is $\langle\tilde Q_5\rangle_{\natural}=1$ and $\Theta_{\tilde Q_5}$ is the half-turn about the line $\mathbb{C}\vec e_z$: it fixes $\vec e_z$ and negates $\vec e_x,\vec e_y$, of determinant $+1$; indeed $\Theta_{\vec e_x\vec e_y}=\Theta_{\vec e_x}\Theta_{\vec e_y}$ conjugates the two half-turns into a rotation, and here the composition is again a half-turn because the lines are orthogonal.

A null vector. For $\tilde{Q}_4=\vec e_x+i\vec e_y$, $\langle\tilde{Q}_4\rangle_{\natural}=0$: the element is not in the pin group, $\Theta_{\tilde{Q}_4}(e_t)=0$, and the reflection formula does not apply, its normalisation being undefined.

Summary

The pin group of the general quaternionic bilinear form is $\mathrm{Pin}=\{\tilde Q_5\in\mathbb{B}^\times:\langle\tilde Q_5\rangle_{\natural}=\pm1\}$ and the spin group is its norm-one part $\mathrm{Spin}=\{\tilde Q_5:\langle\tilde Q_5\rangle_{\natural}=1\}$, normal of index two with $\mathrm{Pin}/\mathrm{Spin}\cong\{\pm1\}$; in the matrix picture they are $\{\det^{2}=1\}$ and $SL_2(\mathbb{C})$, the spin group being connected of real dimension $6$. The twisted operator $\Theta_{\tilde Q_5}$ is a form-preserving map exactly on $\mathrm{Pin}$, and the assignment is a group homomorphism with kernel $\{\pm e_t\}$, a two-to-one cover of its image in the orthogonal group. On the spin group the operator is the inner automorphism, and the image is $\mathrm{Inn}(\mathbb{B})\cong PGL_2(\mathbb{C})\cong SO_3(\mathbb{C})$, acting by fixing the centre and rotating the vector subspace: the classical two-to-one cover of the rotation group by $SL_2(\mathbb{C})$. The versors are the products of non-isotropic vectors of the vector subspace; the norm multiplies over their factors, so a versor is a nonzero multiple of an element of $\mathrm{Pin}$. The sandwich of every unit preserves the vector subspace and acts on it by a similarity of multiplier $\langle\tilde Q_5\rangle_{\natural}^{2}$, so that on the pin group it is a form-preserving map; over this algebra the Clifford group is the whole unit group, because $\mathbb{B}\cong M_2(\mathbb{C})$ has only inner automorphisms, and the general warning that a norm-one unit may fail to preserve the space of vectors does not bite. A single vector carries a half-turn up to the homothety of its norm, and a vector of norm minus one carries the honest reflection. The norm-one vectors generate $\mathrm{Spin}$, the products of two of them the rotations, and by Cartan–Dieudonné every rotation of the vector subspace is such a product; the norm-minus-one coset supplies the automorphisms that are multiplicative only up to the sign, which is what makes the pin cover larger than the spin one.

Summary of Notation

Further Reading

• Two-Sided Operators on the Biquaternion Algebra, §The Isometry Group and the Absence of a Pin Group, the plain sibling, where the shell $N=\pm1$ has no counterpart, the isometry condition is vacuous and the reflections of the plain form lie outside the two-sided family.
• Two-Sided Operators on the General Quaternionic Algebra, for the operator and its laws
• One-Sided Operators on the General Quaternionic Algebra, for the factors and the multiplicative criterion
• The Biquaternion Unit Group as a Topological Group, for the ambient group and its topology
• The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra, for the form and its orthogonal group
• The Clifford, Pin and Spin Groups with Signed Inner Conjugation, for the general theory
• Versors, Rotors and the Sandwich Action with Signed Inner Conjugation, for the general versor and sandwich theory that the versor section specialises