Quaternion Roots of Minus One

Introduction

The equation $\xi^2 = -1$ has, in the quaternion algebra, not two solutions but a whole two-dimensional family: the pure unit quaternions. This article determines that family, describes it as a single conjugacy class and as a homogeneous space of the unit group, relates it to the complex structures that the algebra places on its underlying real vector space, solves the companion equation $\xi^2 = +1$, and compares the answer with the biquaternion and split-quaternion cases. It is the article of the quaternion family that owns the square roots of $-1$; the unit group itself is from Quaternion Norm and Invertibility, and the adjoint action that realises the conjugacy class is from Quaternion Rotations and Reflections.

The single fact on which everything rests is the absence of zero divisors: since a product in $\mathbb{H}$ vanishes only when a factor vanishes, the equation $(\tilde q-1)(\tilde q+1) = 0$ forces $\tilde q = \pm 1$, and a companion argument forces the roots of $-1$ to be pure. Both the smallness of the root set of $+1$ and the roundness of the root set of $-1$ are consequences of the same property, and both fail as soon as the coefficient field is enlarged.

The corpus's default base is a commutative ring with identity, and the algebra is defined there by its presentation. The results below are stated over $\mathbb{R}$, where the algebra is a division algebra; the reduction of the equation and the classification of the roots hold over any field of characteristic not $2$ in which the algebra is a division algebra, and the identification of the root sets with level sets of the central product is the statement over $\mathbb{R}$.

Throughout, $\mathbb{H}$ is the quaternion algebra with basis $e_0 = 1, e_1, e_2, e_3$ and $e_k^2 = -e_0$, a quaternion is $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with scalar part $q_0$ and vector part $\mathbf{q}\in\operatorname{Im}\mathbb{H}$, the conjugate is $\tilde{q}^{\natural} = q_0 e_0 - q_1 e_1 - q_2 e_2 - q_3 e_3$, and the quaternion norm $N$, the unit group $\mathbb{H}^{\times} = \mathbb{H}\setminus\{0\}$ and the unit sphere $Sp(1) = S^3$ are those of Quaternion Norm and Invertibility, named here and not defined again.

The Problem and Its Reduction

Statement

Theorem. A quaternion $\tilde q\in\mathbb{H}$ satisfies $\tilde q^2 = -1$ if and only if $\tilde q$ is a pure imaginary quaternion with $N = 1$,

$$ \tilde q\in\operatorname{Im}\mathbb{H}\cap Sp(1) = \{\tilde q\in\operatorname{Im}\mathbb{H} : N(\tilde q) = 1\}. $$

Proof. Write $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with $q_0\in\mathbb{R}$ and $\mathbf{q}\in\operatorname{Im}\mathbb{H}$. The square is

$$ \tilde q^2 = q_0^2 - N(\mathbf{q}) + 2q_0\mathbf{q}, $$

because the square of a pure imaginary is the real number $\mathbf{q}^2 = -N(\mathbf{q})$, as in Quaternion Norm and Invertibility. Hence $\tilde q^2 = -1$ is the pair of equations

$$ 2q_0\mathbf{q} = 0, \qquad q_0^2 - N(\mathbf{q}) = -1 . $$

If $q_0\neq 0$ the first equation gives $\mathbf{q} = 0$, and the second gives $q_0^2 = -1$, which has no real solution. Therefore $q_0 = 0$, and the second equation becomes $N(\mathbf{q}) = 1$. Conversely a pure imaginary element with $N = 1$ plainly has square $-N(\mathbf{q}) = -1$.

Vector-Part Decomposition

The reduction shows that the square of a quaternion separates into a scalar equation and a vector equation, with the vector equation linear in $\mathbf{q}$ and the scalar equation quadratic. The scalar equation is an equation for the scalar part and the central product $N(\mathbf{q})$ of the vector part together; the vector equation constrains the direction of $\mathbf{q}$ relative to $q_0$.

The Pure Roots

Corollary. The solutions of $\xi^2 = -1$ in $\mathbb{H}$ are exactly the unit vectors of the imaginary subspace.

Proof. By the theorem the solutions are precisely the elements of $\operatorname{Im}\mathbb{H}$ with $N = 1$.

Proposition. The solution set $\Sigma = \{\xi\in\mathbb{H} : \xi^2 = -1\}$ is the set $\operatorname{Im}\mathbb{H}\cap Sp(1)$ of elements of the imaginary subspace with $N = 1$; it has real dimension two in $\mathbb{H}\cong\mathbb{R}^4$, is contained in the level set $\{N = 1\}$, and its topological properties are those of Quaternion Topology.

Proof. The imaginary subspace is a three-dimensional real vector subspace of $\mathbb{H}$ and the solutions are its elements with $N = 1$, so $\Sigma$ has real dimension two; that the level set $\{N = 1\}$ of a three-dimensional real space has dimension two is the standard identification quoted from Quaternion Norm and Invertibility.

Thus the quaternion algebra has a two-dimensional family of square roots of $-1$, where the complex numbers have two, $\pm i$. The excess is not an accident of notation: it is the same excess that makes the quaternion algebra non-commutative, and it is the reason the exponential and the polar form are many-to-one on the real points of $Sp(1)$.

The Classification

Statement of the Theorem

Theorem. The root set $\Sigma = \{\xi : \xi^2 = -1\}$ is a single conjugacy class of the unit group $\mathbb{H}^{\times}$ under the adjoint action, and the map $\tilde q\mapsto \tilde q e_1\tilde q^{-1}$ is a surjection $Sp(1)\to\Sigma$ with fibres the cosets of the stabiliser $U(1)$ of $e_1$.

Proof. The adjoint action $\operatorname{Ad}_{\tilde q}(\tilde p) = \tilde q\tilde p\tilde q^{-1}$ of a unit quaternion preserves the imaginary subspace and the quaternion norm, hence carries $\Sigma$ to itself, and it is transitive on $\Sigma$: the transitivity of the adjoint action on the imaginary subspace is in Quaternion Rotations and Reflections. Hence $\Sigma$ is one orbit, that is, one conjugacy class. The stabiliser of $e_1$ is the set of units with $\tilde q e_1\tilde q^{-1} = e_1$, namely the units commuting with $e_1$; the commutant of $e_1$ in $\mathbb{H}$ is the two-dimensional plane $\mathbb{R}\oplus\mathbb{R}e_1$, whose unit circle is $U(1) = \{e^{e_1\theta}\}$. The orbit is therefore $Sp(1)/U(1)$ as a set with a transitive action, and the fibres of the orbit map are the cosets of $U(1)$.

The Homogeneous Space

Theorem. There is a canonical bijection of homogeneous spaces

$$ \Sigma \cong Sp(1)/U(1) \cong SO(3)/SO(2), $$

and under these identifications $\Sigma$ has real dimension two.

Proof. The orbit–stabiliser theorem gives $\Sigma\cong Sp(1)/U(1)$ as a homogeneous space of $Sp(1)$, and the identification $Sp(1)/\{\pm1\}\cong SO(3)$ of Quaternion Rotations and Reflections identifies it with $SO(3)/SO(2)$ because $U(1)$ contains $\{\pm1\}$. That $SO(3)/SO(2)$ has real dimension two, and with it the identification of $\Sigma$ with $SO(3)/SO(2)$, is the classical statement quoted from Quaternion Topology.

Verification

The statement can be checked in coordinates. A root is $\xi = u_1e_1+u_2e_2+u_3e_3$ with $u_1^2+u_2^2+u_3^2 = 1$, and conjugating $e_1$ by the unit quaternion $\tilde q = q_0+q_1e_1+q_2e_2+q_3e_3$ produces the root $\tilde q e_1\tilde q^{-1}$, whose coordinates are the entries of the matrix of the adjoint action applied to $e_1$; as $\tilde q$ ranges over $Sp(1)$ these conjugate roots fill the whole of $\Sigma$, by the transitivity quoted above. So the two descriptions of $\Sigma$ agree.

Status of the Roots

Each root $\xi\in\Sigma$ is a unit quaternion, hence invertible, with $\xi^{-1} = \bar\xi = -\xi$. It generates a subalgebra $\mathbb{R}\oplus\mathbb{R}\xi\cong\mathbb{C}$ inside $\mathbb{H}$, and the square of $\xi$ is the scalar $-1$; this is the sense in which each root gives a copy of the complex numbers inside the quaternions. Distinct roots $\xi\neq\eta$ generate distinct, but conjugate, copies: there is a unit quaternion carrying one onto the other, although no quaternion commutes with two distinct roots except a real number.

The Relation to the Complex Structures on $\mathbb{R}^4$

Definition. A complex structure on the real vector space $\mathbb{H}$ is a real-linear map $J : \mathbb{H}\to\mathbb{H}$ with $J^2 = -\mathrm{id}$. A complex structure $J$ is compatible with the quaternion structure if it is left multiplication by a root, $J = L_\xi$ with $\xi^2 = -1$, where $L_\xi(\tilde q) = \xi\tilde q$.

Theorem. For every $\xi\in\Sigma$ the operator $L_\xi$ of left multiplication by $\xi$ is a complex structure on $\mathbb{H}$, and the assignment $\xi\mapsto L_\xi$ is a bijection from $\Sigma$ onto the set of complex structures on $\mathbb{H}$ compatible with the quaternion structure.

Proof. Left multiplication is real-linear, and $L_\xi^2 = L_{\xi^2} = L_{-1} = -\mathrm{id}$ because $\xi^2 = -1$. Conversely, if $L_\xi = L_\eta$ then $\xi = L_\xi(1) = L_\eta(1) = \eta$, so the map is injective.

Proposition. The three elements $e_1,e_2,e_3$ give three complex structures $I_1,I_2,I_3$ satisfying the quaternion relations

$$ I_1^2 = I_2^2 = I_3^2 = -1, \qquad I_1I_2 = I_3, \quad I_2I_3 = I_1, \quad I_3I_1 = I_2 , $$

and the whole two-dimensional set $\Sigma$ of roots is the family of complex structures spanned by them.

Proof. The relations are the images of the algebra relations $e_k^2 = -e_0$ and $e_1e_2 = e_3$ under the injective algebra map $L$. A general root is $\xi = u_1e_1+u_2e_2+u_3e_3$ with $u_1^2+u_2^2+u_3^2 = 1$, and $L_\xi = u_1I_1+u_2I_2+u_3I_3$.

The family $\{I_1,I_2,I_3\}$ is a quaternionic structure on $\mathbb{H}$: a triple of anticommuting complex structures, whose span is the two-dimensional set $\Sigma$, and whose existence is the same statement as the non-commutativity of the algebra. The complex line spanned by $1$ and $\xi$ is a complex structure in the algebraic sense — a subalgebra isomorphic to $\mathbb{C}$ — while $L_\xi$ is a complex structure in the linear sense.

The Roots of Plus One

Theorem. The solutions of $\eta^2 = 1$ in $\mathbb{H}$ are exactly $\eta = \pm 1$.

Proof. The identity $\eta^2-1 = (\eta-1)(\eta+1)$ holds by distributivity, and $\mathbb{H}$ has no zero divisors, so $\eta^2 = 1$ implies $\eta-1 = 0$ or $\eta+1 = 0$. The two values both satisfy the equation.

Corollary. The set of roots of $+1$ has two elements, and it is the centre of the group $\{N = 1\}$: $\{\pm1\} = Z(Sp(1))$.

Proof. The centre of $Sp(1)$ consists of the units commuting with every quaternion, which are the elements of the centre $\mathbb{R}$ with $N = 1$, namely $\pm1$.

Remark. The contrast between the two equations is complete. The root set of $+1$ is discrete and realizes the centre; the root set of $-1$ has real dimension two and realizes a single conjugacy class, the class of $e_1$, and it is as far from central as a conjugacy class can be: only the trivial class is smaller in the quotient. The equation $\eta^2 = 1$ has as many solutions as an algebra with no zero divisors permits — exactly the two central ones — and it has no solution outside $\mathbb{R}$.

Comparison with the Biquaternion and Split-Quaternion Cases

The root set of $-1$ grows as soon as the coefficient field is enlarged, and the enlargement is the measure of how far the algebra has moved from the division case.

Proposition. In the split quaternion algebra $\mathbb{H}_{\mathrm{s}}$, with basis $1,e_1,e_2,e_3$, $e_1^2 = -1$, $e_2^2 = +1$, $e_3 = e_1e_2$, the solutions of $\xi^2 = -1$ are the points of the level set $N = 1$

$$ q_1^2 - q_2^2 - q_3^2 = 1, \qquad \xi = q_1e_1+q_2e_2+q_3e_3 , $$

a set of dimension two rather than a surface of higher dimension.

The derivation is in Split-Quaternion Roots of Minus One, where the split quaternion algebra is treated; only the comparison is recorded here.

Algebra	Roots of $-1$	Roots of $+1$
$\mathbb{H}$	the pure elements with $N = 1$	$\{\pm1\}$
$\mathbb{H}_{\mathrm{s}}$	the level set $q_1^2-q_2^2-q_3^2 = 1$ (in $q_0 = 0$)	the level set $q_2^2+q_3^2-q_1^2 = 1$ (in $q_0 = 0$), together with $\pm1$
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	a two-dimensional family together with further complex sheets	the roots $\xi i$ obtained from those of $-1$, including $\pm\mu i$ for a pure unit $\mu$

The biquaternion root set is computed in Biquaternion Square Roots of Minus One, Zero and Plus One; it contains the pure roots but also roots with complex coordinates, from which the non-trivial idempotents $\tfrac{1}{2}(1+\xi i)$ are built. The presence of those extra roots is the failure of division: in a division algebra the factorisation argument of the roots-of-$+1$ theorem is available and the root set is small, while in $\mathbb{H}_{\mathrm{s}}$ and in $\mathbb{B}$ the same factorisation fails.

Summary

A quaternion satisfies $\xi^2 = -1$ exactly when it is a pure imaginary quaternion with $N = 1$, so the root set is $\Sigma = \operatorname{Im}\mathbb{H}\cap Sp(1)$, the level set $\{N = 1\}$, a two-dimensional family in place of the two roots $\pm i$ of the complex case. The square of a general quaternion splits into the scalar equation $q_0^2-N(\mathbf{q}) = -1$ and the vector equation $2q_0\mathbf{q} = 0$, and the pair forces $q_0 = 0$ and $N(\mathbf{q}) = 1$.

The root set is a single conjugacy class of the unit group under the adjoint action, and the orbit–stabiliser theorem identifies it with the homogeneous space $Sp(1)/U(1)\cong SO(3)/SO(2)$, the stabiliser of $e_1$ being the circle of units commuting with $e_1$. Each root gives a complex structure $L_\xi$ on the underlying real space $\mathbb{H}\cong\mathbb{R}^4$, and the three roots $e_1,e_2,e_3$ give the anticommuting triple $I_1,I_2,I_3$ of a quaternionic structure whose span is the whole two-dimensional set of roots.

The companion equation $\eta^2 = 1$ has only the two solutions $\pm1$, because the factorisation $(\eta-1)(\eta+1) = 0$ together with the absence of zero divisors forces $\eta$ to be central; the root set of $+1$ is the centre of $Sp(1)$. In the split quaternion algebra the roots of $-1$ fill a set with two components rather than one, and in the biquaternion algebra they include further complex sheets, from which the non-trivial idempotents are built; in both cases the extra roots are bought with the loss of the division property.

Summary of Notation

Symbol	Meaning
$\mathbb{H}$	The quaternion algebra
$e_0 = 1, e_1, e_2, e_3$	Quaternion basis, $e_k^2 = -e_0$
$\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$	General quaternion, scalar part $q_0$, vector part $\mathbf{q}$
$\tilde{q}^{\natural} = q_0 e_0 - q_1 e_1 - q_2 e_2 - q_3 e_3$	Quaternion conjugate
$N$	Quaternion norm, from Quaternion Norm and Invertibility
$\operatorname{Im}\mathbb{H}$	Pure imaginary quaternions, $\cong\mathbb{R}^3$
$\Sigma = \{\xi : \xi^2 = -1\} = \operatorname{Im}\mathbb{H}\cap Sp(1)$	Root set, the level set $\{N = 1\}$ in $\operatorname{Im}\mathbb{H}$
$Sp(1) = \{N = 1\}$	The group $\{N = 1\}$ of units, from Quaternion Norm and Invertibility
$U(1) = \{e^{e_1\theta}\}$	Stabiliser of $e_1$, the commutant unit circle
$\Sigma\cong Sp(1)/U(1)\cong SO(3)/SO(2)$	Root set as a homogeneous space
$L_\xi(\tilde q) = \xi\tilde q$	Complex structure of left multiplication by a root
$I_1,I_2,I_3$	Complex structures $L_{e_1},L_{e_2},L_{e_3}$, a quaternionic structure
$\mathbb{H}_{\mathrm{s}}$	Split quaternion algebra, for contrast
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	Biquaternion algebra, for contrast