Square Roots of One and the Idempotents of the Biquaternion Algebra

Introduction

Of the four general products of The Four General Products and Their Relations over the Biquaternion Algebra, exactly one makes the underlying complex space of $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ an associative unital $\mathbb{C}$-algebra; the algebra is The Biquaternion Algebra and Its Vector Spaces, and its multiplication is the general plain bilinear product of two biquaternions, $\tilde Q_1\tilde Q_2$, the product being the one of the algebra. That product is the one read throughout this article.

The article treats two subjects, and in this order. The first is the square roots of the central value $+1$, the elements $\tilde\Xi\in\mathbb{B}$ with

$$ \tilde\Xi^2=e_t . $$

The second is the idempotents of the algebra, the elements $\tilde\Pi$ with $\tilde\Pi^2=\tilde\Pi$. The two subjects are one subject, because the halving map

$$ H(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi),\qquad \tilde\Xi^2=e_t, $$

is a bijection from the roots of $+1$ onto the idempotents, with inverse $H^{-1}(\tilde\Pi)=2\tilde\Pi-e_t$. A root carries two idempotents and not one: the halving map picks the member attached to $+1$, and the other is its value at $-\tilde\Xi$; with $\tilde\Pi_{+}(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi)$ and $\tilde\Pi_{-}(\tilde\Xi)=\tfrac12(e_t-\tilde\Xi)=\tilde\Pi_{+}(-\tilde\Xi)$, the two are a complete orthogonal pair, $\tilde\Pi_{+}(\tilde\Xi)+\tilde\Pi_{-}(\tilde\Xi)=e_t$ and $\tilde\Pi_{+}(\tilde\Xi)\tilde\Pi_{-}(\tilde\Xi)=0$, and the root is their difference, $\tilde\Xi=\tilde\Pi_{+}(\tilde\Xi)-\tilde\Pi_{-}(\tilde\Xi)$. The reason is a two-line computation: if $\tilde\Xi^2=e_t$ then $\tilde\Pi^2=\tfrac14(e_t+2\tilde\Xi+e_t)=\tilde\Pi$, and if $\tilde\Pi^2=\tilde\Pi$ then $(2\tilde\Pi-e_t)^2=4\tilde\Pi^2-4\tilde\Pi+e_t=e_t$. Every idempotent of the algebra is read through that map, on the roots of $+1$ alone, and the families of idempotents are the families of the roots of $+1$ read through it.

The results. The roots of $+1$ are the two trivial roots $\pm e_t$; the purely imaginary roots $\pm i\vec u$ over the real unit vectors $\vec u$ with $\vec u^2=-e_t$, a two-real-parameter family; and the non-Hermitian roots $\vec\xi+i\vec\xi'$ with $\vec\xi\neq0$ and the two real conditions $\xi_x^2+\xi_y^2+\xi_z^2-(\xi'_x)^2-(\xi'_y)^2-(\xi'_z)^2=-1$ and $\xi_x\xi'_x+\xi_y\xi'_y+\xi_z\xi'_z=0$, a four-real-parameter family. There are no others. Every idempotent of the algebra is trivial, $0$ or $e_t$, or of the form $\tilde\Pi=\tfrac12(e_t+\tilde\Xi)$ with $\tilde\Xi$ a root of $+1$; the trivial roots give the trivial idempotents, the purely imaginary roots give the non-trivial Hermitian idempotents, and the non-Hermitian roots give a four-real-parameter family of non-Hermitian idempotents lying in none of the four four-dimensional remarkable subspaces. Each root carries the two idempotents $\tilde\Pi_{+}(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi)$ and $\tilde\Pi_{-}(\tilde\Xi)=\tfrac12(e_t-\tilde\Xi)=e_t-\tilde\Pi_{+}(\tilde\Xi)$, a complete orthogonal pair whose difference is the root; the class $\{\tilde\Xi,-\tilde\Xi\}$ gives the same unordered pair.

The following table collects the results at a glance.

The article is algebraic. It does not treat the roots of the general quadratic $\tilde\Xi^2=\tilde Q$, which is Square Roots of a General Element in the Biquaternion Algebra; the roots of $0$ and the nilpotents, which are Square Roots of Zero and the Nilpotents of the Biquaternion Algebra; the structure of the zero-divisor set, which is Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors; or the ideals the standard idempotents generate, which are Ideals and Peirce Decompositions of the Biquaternion Algebra. The subsection §The Roots of Minus One records how the roots of $-1$ are read off the roots of $+1$; it is a remark, it is used nowhere, and it opens nothing.

$$ (\vec Q_1,\vec Q_2)=\sum_kQ_{1k}Q_{2k}, $$

and the square of a pure element is $\vec Q^2=-(\vec Q,\vec Q)e_t$. The remarkable subspaces $\mathbb{C}_{\mathbb{B}}$, $\mathrm{Vect}(\mathbb{B})$, $\mathbb{H}_{\mathbb{B}}$, $i\mathbb{H}_{\mathbb{B}}$, $\mathbb{M}_+$ and $\mathbb{M}_-$ are Introduction to the Remarkable Subspaces. A root of a central value is written $\tilde\Xi$ (§The Root Convention of Conventions in Mathematics), an idempotent $\tilde\Pi$ (§The Idempotent Convention), the two members of a complete pair with the two signs, $\tilde\Pi_{+}$ and $\tilde\Pi_{-}$, the subscript $+$ naming the member attached to $+1$, and a generic element $\tilde Q$.

The equations name a product, so the roots and the idempotents are relative to it. The general definition of the power, the square and the root of a central value is Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, §Powers, the Square and the Roots, and the general definition of an idempotent, with the orthogonal, complete and primitive families, the Hermitian idempotent and the Peirce decomposition, is the same article. Throughout this article the square is the one of the general plain bilinear product, the multiplication of the algebra; the other three general products carry roots and idempotents of their own, and they are Square Roots of One and the Idempotents of the General Plain Sesqualgebra, Square Roots of One and the Idempotents of the General Quaternionic Algebra and Square Roots of One and the Idempotents of the General Quaternionic Sesqualgebra.

The Roots of One

The roots of $+1$ are exactly the trivial roots $\pm e_t$, the purely imaginary roots $\pm i\vec u$, and the non-Hermitian roots $\vec\xi+i\vec\xi'$; there are no others.

A root of $+1$ is an element $\tilde\Xi\in\mathbb{B}$ with $\tilde\Xi^2=e_t$, the square being the one of the general plain bilinear product.

The Reduction

Write the root in scalar and vector parts,

$$ \tilde\Xi=\Xi_te_t+\vec\Xi,\qquad \vec\Xi=\Xi_x\vec e_x+\Xi_y\vec e_y+\Xi_z\vec e_z,\qquad \Xi_k\in\mathbb{C}, $$

and write the complex coefficients as $\Xi_k=\xi_k+i\xi'_k$ with real $\xi_k,\xi'_k$, so that the vector part splits into two real vector parts,

$$ \vec\Xi=\vec\xi+i\vec\xi',\qquad \vec\xi=\xi_x\vec e_x+\xi_y\vec e_y+\xi_z\vec e_z,\qquad \vec\xi'=\xi'_x\vec e_x+\xi'_y\vec e_y+\xi'_z\vec e_z . $$

The square of a biquaternion is given by the product formula,

$$ \tilde\Xi^2=\bigl(\Xi_t^2-(\vec\Xi,\vec\Xi)\bigr)e_t+2\Xi_t\vec\Xi,\qquad (\vec\Xi,\vec\Xi)=\Xi_x^2+\Xi_y^2+\Xi_z^2, $$

so equating $\tilde\Xi^2$ to $e_t$ splits the equation into a vector part and a scalar part,

$$ 2\Xi_t\vec\Xi=0, $$

$$ \Xi_t^2-(\vec\Xi,\vec\Xi)=1 . $$

Since $\mathbb{C}$ is a field, the vector equation holds exactly when $\Xi_t=0$ or $\vec\Xi=0$. The roots of $+1$ therefore split into two disjoint cases: the root is a complex scalar, $\vec\Xi=0$, or the root is pure, $\Xi_t=0$. The split is collected in the table.

The Trivial Roots

In the scalar case $\tilde\Xi=\Xi_te_t$ and the scalar equation reads $\Xi_t^2=1$, so $\Xi_t=\pm1$. The roots in this case are the two elements

$$ \tilde\Xi=+e_t,\qquad \tilde\Xi=-e_t, $$

the trivial roots of $+1$. They are the only roots of $+1$ in the centre $\mathbb{C}e_t$, and the only ones with non-vanishing scalar part.

The Non-Trivial Roots

The non-trivial roots are the pure roots, $\tilde\Xi=\vec\Xi$ with $(\vec\Xi,\vec\Xi)=-1$; they split into the purely imaginary roots and the non-Hermitian roots.

The pure case splits by whether the vector part is $i$ times a real vector. The first family is the purely imaginary roots; the second is the non-Hermitian roots.

The Purely Imaginary Roots

In the pure case $\tilde\Xi=\vec\Xi$ and the scalar equation reads

$$ (\vec\Xi,\vec\Xi)=\Xi_x^2+\Xi_y^2+\Xi_z^2=-1 . $$

Take first the case in which the vector part is the central imaginary times a real vector, $\vec\Xi=i\vec u$ with $\vec u$ a unit real vector. Since $i$ is central and $\vec u^2=-\|\vec u\|^2e_t$ with $\|\vec u\|^2=u_x^2+u_y^2+u_z^2$, one has

$$ (i\vec u)^2=i^2\vec u^2=(-1)\bigl(-\|\vec u\|^2e_t\bigr)=\|\vec u\|^2e_t, $$

so such an element is a root of $+1$ exactly when $\|\vec u\|^2=1$, that is when $\vec u^2=-e_t$. The roots in this case are the elements

$$ \tilde\Xi=+i\vec u,\qquad \tilde\Xi=-i\vec u,\qquad \vec u\ \text{a unit real vector with}\ \vec u^2=-e_t, $$

a two-real-parameter family: three real coefficients of $\vec u$ carrying the single condition $u_x^2+u_y^2+u_z^2=1$. Its members are the only roots of $+1$ whose vector part is $i$ times a real vector, and each is Hermitian, $(\tilde\Xi)^{*}=\tilde\Xi$.

The Non-Hermitian Roots

Let now the vector part be genuinely complex, $\vec\Xi=\vec\xi+i\vec\xi'$ with $\vec\xi,\vec\xi'$ real pure. Then

$$ (\vec\Xi,\vec\Xi)=\sum_k(\xi_k+i\xi'_k)^2=\bigl(\xi_x^2+\xi_y^2+\xi_z^2-(\xi'_x)^2-(\xi'_y)^2-(\xi'_z)^2\bigr)+2i\bigl(\xi_x\xi'_x+\xi_y\xi'_y+\xi_z\xi'_z\bigr), $$

so the single complex equation $(\vec\Xi,\vec\Xi)=-1$ is the two real equations

$$ \xi_x^2+\xi_y^2+\xi_z^2-(\xi'_x)^2-(\xi'_y)^2-(\xi'_z)^2=-1,\qquad \xi_x\xi'_x+\xi_y\xi'_y+\xi_z\xi'_z=0 . $$

The case $\vec\xi=0$ requires $(\xi'_x)^2+(\xi'_y)^2+(\xi'_z)^2=1$ and is the two-real-parameter family of the previous paragraph. The remaining case is $\vec\xi\neq0$: six real coefficients carrying the two constraints, hence a four-real-parameter family. Its elements are the non-Hermitian roots of $+1$, the roots that are not fixed by the star, $\tilde\Xi^{*}\neq\tilde\Xi$,

$$ \tilde\Xi=\vec\xi+i\vec\xi',\qquad \vec\xi\neq0, $$

$$ \xi_x^2+\xi_y^2+\xi_z^2-(\xi'_x)^2-(\xi'_y)^2-(\xi'_z)^2=-1,\qquad \xi_x\xi'_x+\xi_y\xi'_y+\xi_z\xi'_z=0, $$

equivalently $\vec\xi\vec\xi'+\vec\xi'\vec\xi=0$. For such a root the imaginary part is automatically nonzero: if $\vec\xi'=0$ the first equation would give $\xi_x^2+\xi_y^2+\xi_z^2=-1$, impossible, the three squares being non-negative.

Statement of the Theorem

Theorem (the roots of $+1$). The roots of $+1$ in $\mathbb{B}$ are exactly the elements of the three following families.

The trivial roots. The two central elements

$$ \tilde\Xi=+e_t,\qquad \tilde\Xi=-e_t . $$

The purely imaginary roots. The two-real-parameter family

$$ \boxed{\; \tilde\Xi=+i\vec u,\qquad \tilde\Xi=-i\vec u,\qquad \vec u=u_x\vec e_x+u_y\vec e_y+u_z\vec e_z,\ u_k\in\mathbb{R},\ u_x^2+u_y^2+u_z^2=1 \;}, $$

equivalently over the real unit vectors with $\vec u^2=-e_t$; since $i$ is central the family is also written $\tilde\Xi=\pm\vec u i$.

The non-Hermitian roots. The four-real-parameter family

$$ \tilde\Xi=\vec\xi+i\vec\xi'=\bigl(\xi_x\vec e_x+\xi_y\vec e_y+\xi_z\vec e_z\bigr)+i\bigl(\xi'_x\vec e_x+\xi'_y\vec e_y+\xi'_z\vec e_z\bigr),\qquad \vec\xi\neq0, $$

whose six real coefficients satisfy the two conditions

$$ \xi_x^2+\xi_y^2+\xi_z^2-(\xi'_x)^2-(\xi'_y)^2-(\xi'_z)^2=-1,\qquad \xi_x\xi'_x+\xi_y\xi'_y+\xi_z\xi'_z=0, $$

equivalently $\vec\xi\vec\xi'+\vec\xi'\vec\xi=0$.

The three families are disjoint, and together they are all the roots.

Proof. The reduction splits the equation into $2\Xi_t\vec\Xi=0$ and $\Xi_t^2-(\vec\Xi,\vec\Xi)=1$. If $\vec\Xi=0$ then $\Xi_t=\pm1$, the two trivial roots. If $\Xi_t=0$ then $(\vec\Xi,\vec\Xi)=-1$; writing $\vec\Xi=\vec\xi+i\vec\xi'$ with real $\vec\xi,\vec\xi'$ turns this into the two real equations of the theorem. The case $\vec\xi=0$ is the purely imaginary family, and the case $\vec\xi\neq0$ is the non-Hermitian family. The cases $\vec\Xi=0$ and $\Xi_t=0$ are disjoint and exhaustive by the reduction, and within the pure case the two subcases are disjoint and exhaustive; hence the list is complete.

The roots fall into the three following families.

Verification

Each family consists of roots of $+1$.

Trivial roots. $(\pm e_t)^2=e_t$.

Purely imaginary roots. For $\vec u^2=-e_t$, $(i\vec u)^2=i^2\vec u^2=(-1)(-e_t)=e_t$.

Non-Hermitian roots. Expanding,

$$ \tilde\Xi^2=(\vec\xi+i\vec\xi')^2=\vec\xi^2-(\vec\xi')^2+i\bigl(\vec\xi\vec\xi'+\vec\xi'\vec\xi\bigr). $$

Now $\vec\xi^2=-\bigl(\xi_x^2+\xi_y^2+\xi_z^2\bigr)e_t$, $(\vec\xi')^2=-\bigl((\xi'_x)^2+(\xi'_y)^2+(\xi'_z)^2\bigr)e_t$, and the mixed term is a central multiple,

$$ \vec\xi\vec\xi'+\vec\xi'\vec\xi=-2\bigl(\xi_x\xi'_x+\xi_y\xi'_y+\xi_z\xi'_z\bigr)e_t=0, $$

by the second condition. Hence

$$ \tilde\Xi^2=-\bigl(\xi_x^2+\xi_y^2+\xi_z^2-(\xi'_x)^2-(\xi'_y)^2-(\xi'_z)^2\bigr)e_t=-(-1)e_t=e_t, $$

by the first condition.

The three checks are collected in the table.

The Roots of Minus One

Proposition (the roots of $-1$). The equation $\tilde\Xi^{2}=-e_t$ has the two central scalars $\pm i$; the family $\pm\vec u$ over the real unit vectors $\vec u$ with $\vec u^2=-e_t$; and the family $-\vec\xi'+i\vec\xi$ with the same two real conditions as the non-Hermitian roots of $+1$, with the sign of the first reversed.

Proof. The central imaginary turns the roots of $+1$ into the roots of $-1$: if $\tilde\Xi^2=e_t$ then $(i\tilde\Xi)^2=i^2\tilde\Xi^2=-e_t$, because $i$ is central, and since $i$ is invertible the map $\tilde\Xi\mapsto i\tilde\Xi$ is a bijection from the roots of $+1$ onto the roots of $-1$. The trivial roots go to $\pm i$, the purely imaginary roots to $\pm\vec u$, and the non-Hermitian roots to $-\vec\xi'+i\vec\xi$.

The result is collected in the table.

The mechanism is not universal. It needs a central scalar of square $-1$ that can be moved past the two slots of the product; in a sesquilinear product, where the second slot conjugates, scalar multiplication cannot flip the sign this way, and the roots of $-1$ are then read from the roots of $+1$ by no such map, or are absent altogether. The roots of $-1$ are quoted here and are used nowhere in this article.

Status of the Roots

Proposition (the roots are not closed under multiplication). The product of two roots of $+1$ need not be a root of $+1$: the purely imaginary roots $i\vec e_x$ and $i\vec e_y$ are roots of $+1$, but

$$ (i\vec e_x)(i\vec e_y)=i^{2}\vec e_x\vec e_y=-\vec e_z $$

is a unit and not a root of $+1$. Hence the roots form no group.

Proof. $(i\vec e_x)^{2}=(i\vec e_y)^{2}=e_t$, because $i^{2}=-1$ and $\vec e_x^{2}=\vec e_y^{2}=-e_t$; their product is $-\vec e_z$, whose square is $\vec e_z^{2}=-e_t\neq e_t$, and which is a unit with inverse $\vec e_z$.

Proposition (the roots are units). Every root of $+1$ is a unit, with $\tilde\Xi^{-1}=\tilde\Xi$, and no root is a zero divisor.

Proof. $\tilde\Xi^2=e_t$ reads $\tilde\Xi\tilde\Xi=e_t$, so $\tilde\Xi$ is invertible with inverse $\tilde\Xi$. A unit is not a zero divisor.

Proposition (purity and the common element). Every root except the trivial ones is pure, and the only element common to the roots of $+1$ and the idempotents is $e_t$.

Proof. The reduction shows that a root is either a complex scalar, hence one of $\pm e_t$, or pure. If $\tilde\Xi$ is both a root of $+1$ and an idempotent, then $\tilde\Xi^2=e_t$ and $\tilde\Xi^2=\tilde\Xi$, hence $\tilde\Xi=e_t$.

Counting the free real parameters: the trivial roots are two isolated elements, the purely imaginary roots are a two-real-parameter family, and the non-Hermitian roots are a four-real-parameter family.

The non-trivial roots therefore lie in the six-real-dimensional vector subspace of pure biquaternions, while the trivial roots lie in the centre. The status of the roots is collected in the table.

The Idempotents

The idempotents of $\mathbb{B}$ are exactly the trivial idempotents $0,e_t$, the non-trivial Hermitian idempotents $\tfrac12(e_t\pm i\vec u)$, and the non-Hermitian idempotents $\tfrac12(e_t+\vec\xi+i\vec\xi')$; the standard pair is orthogonal, complete and primitive, and every idempotent is the image of exactly one root of $+1$ under the halving map.

The idempotents of $\mathbb{B}$ are the elements satisfying $\tilde\Pi^2=\tilde\Pi$, the square being the one of the general plain bilinear product. They correspond to projectors under multiplication, and in $\mathbb{B}$ they do four separate jobs at once: they give the direct sum decompositions of the algebra into left ideals, and in particular the two minimal left ideals; they classify the non-pure zero divisors, every one of which is a complex multiple of an idempotent; they are in bijection with the roots of $+1$ through the halving map, each root carrying a complete orthogonal pair, so their classification is the classification of those roots; and they drive the Peirce decomposition of the algebra. The definition and the families are the general ones of Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, read here in the general plain bilinear product.

The Definition

Definition. An element $\tilde\Pi$ of $\mathbb{B}$ is an idempotent of the biquaternion algebra when

$$ \tilde\Pi^2=\tilde\Pi . $$

The element $0$ is an idempotent, and so is the unit $e_t$; the two are the trivial idempotents. The equation names a product, and the notion is relative to it: throughout this article the product is the general plain bilinear product, the multiplication of the algebra, and another product is named where it is meant. The idempotents of the other three general products are not these; each is the subject of its own article, and the four sets are compared in Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.

Two idempotents $\tilde\Pi,\tilde\Pi'$ are orthogonal when $\tilde\Pi\tilde\Pi'=\tilde\Pi'\tilde\Pi=0$; a family is pairwise orthogonal when its members are pairwise orthogonal, and complete when in addition their sum is $e_t$.

Idempotents encode direct summands of the algebra: an idempotent $\tilde\Pi$ and its complement $e_t-\tilde\Pi$ split the underlying $\mathbb{C}$-module,

$$ \mathbb{B}=\mathbb{B}\tilde\Pi\oplus\mathbb{B}(e_t-\tilde\Pi),\qquad \mathbb{B}=\tilde\Pi\mathbb{B}\oplus(e_t-\tilde\Pi)\mathbb{B}, $$

and every such decomposition arises from an idempotent. A primitive idempotent, one that is not a sum of two nonzero orthogonal idempotents, generates a minimal left ideal. The general statements, with the orthogonal, complete and primitive families, the Hermitian idempotent and the Peirce decomposition, are Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space.

The Two Idempotents of a Root

A root $\tilde\Xi$ of $+1$ carries the two idempotents $\tilde\Pi_{+}(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi)$ and $\tilde\Pi_{-}(\tilde\Xi)=\tfrac12(e_t-\tilde\Xi)=e_t-\tilde\Pi_{+}(\tilde\Xi)$; they are complete and orthogonal, and their difference is the root.

Theorem (the two idempotents of a root). For a root $\tilde\Xi$ of $+1$, the elements

$$ \boxed{\; \tilde\Pi_{+}(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi),\qquad \tilde\Pi_{-}(\tilde\Xi)=\tfrac12(e_t-\tilde\Xi) \;} $$

are idempotent, complete and orthogonal; the identities are collected in the table.

Proof. The idempotence is the computation

$$ \tilde\Pi_{+}(\tilde\Xi)^2=\tfrac14(e_t+\tilde\Xi)^2=\tfrac14(e_t+2\tilde\Xi+\tilde\Xi^2)=\tfrac14(2e_t+2\tilde\Xi)=\tilde\Pi_{+}(\tilde\Xi), $$

and the same computation with $\tilde\Xi$ replaced by $-\tilde\Xi$ gives the second identity. The sum is $\tfrac12(e_t+\tilde\Xi)+\tfrac12(e_t-\tilde\Xi)=e_t$. The product is

$$ \tilde\Pi_{+}(\tilde\Xi)\tilde\Pi_{-}(\tilde\Xi)=\tfrac14(e_t+\tilde\Xi)(e_t-\tilde\Xi)=\tfrac14(e_t-\tilde\Xi^2)=0, $$

and equally in the other order. The difference is $\tfrac12(e_t+\tilde\Xi)-\tfrac12(e_t-\tilde\Xi)=\tilde\Xi$.

The pair is written with the two signs rather than with two numbers, as §The Idempotent Convention of Conventions in Mathematics provides for a complete pair: the subscript $+$ names the member attached to $+1$, the subscript $-$ the member attached to $-1$.

The trivial roots give the trivial pairs: the pair of $+e_t$ is $(e_t,0)$ and the pair of $-e_t$ is $(0,e_t)$. Every root therefore carries exactly two idempotents, and the next section computes what the two do together.

The Arithmetic of the Pair

The pair of a root is complete, $\tilde\Pi_{+} + \tilde\Pi_{-}=e_t$; the root is its difference, $\tilde\Pi_{+} - \tilde\Pi_{-}=\tilde\Xi$; the pair is orthogonal, $\tilde\Pi_{+}\tilde\Pi_{-}=\tilde\Pi_{-}\tilde\Pi_{+}=0$; and multiplication by the root is the sign change of the pair, $+1$ on $\tilde\Pi_{+}$ and $-1$ on $\tilde\Pi_{-}$.

Throughout this section $\tilde\Xi$ is a root of $+1$ and $\tilde\Pi_{+}=\tilde\Pi_{+}(\tilde\Xi)$, $\tilde\Pi_{-}=\tilde\Pi_{-}(\tilde\Xi)$ its two idempotents, as in §The Two Idempotents of a Root.

Addition. The pair is complete: the two idempotents sum to the unit,

$$ \tilde\Pi_{+} + \tilde\Pi_{-}=\tfrac12(e_t+\tilde\Xi)+\tfrac12(e_t-\tilde\Xi)=e_t . $$

Subtraction. The two idempotents differ by the root,

$$ \tilde\Pi_{+} - \tilde\Pi_{-}=\tfrac12(e_t+\tilde\Xi)-\tfrac12(e_t-\tilde\Xi)=\tilde\Xi, $$

so that the root is read on the pair as $\tilde\Xi=\tilde\Pi_{+} - \tilde\Pi_{-}$, equivalently $\tilde\Xi=2\tilde\Pi_{+} - e_t$. Each idempotent is the complement of the other, and replacing the root by $-\tilde\Xi$ interchanges them,

$$ \tilde\Pi_{+}(-\tilde\Xi)=\tilde\Pi_{-}(\tilde\Xi),\qquad \tilde\Pi_{-}(-\tilde\Xi)=\tilde\Pi_{+}(\tilde\Xi). $$

Multiplication. The pair is orthogonal,

$$ \tilde\Pi_{+}\tilde\Pi_{-}=\tfrac14(e_t+\tilde\Xi)(e_t-\tilde\Xi)=\tfrac14(e_t-\tilde\Xi^2)=0, $$

and equally $\tilde\Pi_{-}\tilde\Pi_{+}=0$. The four products of the pair are collected in the multiplication table.

so the $\mathbb{C}$-span $\mathbb{C}\tilde\Pi_{+}\oplus\mathbb{C}\tilde\Pi_{-}$ is a commutative subalgebra isomorphic to $\mathbb{C}\times\mathbb{C}$, with the componentwise product $(s\tilde\Pi_{+} + t\tilde\Pi_{-})(s'\tilde\Pi_{+} + t'\tilde\Pi_{-})=ss'\tilde\Pi_{+} + tt'\tilde\Pi_{-}$, and the root is the element $(+1,-1)$ of that algebra.

Multiplication by the root is diagonal on the pair. On the left,

$$ \tilde\Xi\tilde\Pi_{+}=\tfrac12(\tilde\Xi+\tilde\Xi^2)=\tfrac12(e_t+\tilde\Xi)=\tilde\Pi_{+},\qquad \tilde\Xi\tilde\Pi_{-}=\tfrac12(\tilde\Xi-\tilde\Xi^2)=-\tfrac12(e_t-\tilde\Xi)=-\tilde\Pi_{-}, $$

and on the right $\tilde\Pi_{+}\tilde\Xi=\tilde\Pi_{+}$ and $\tilde\Pi_{-}\tilde\Xi=-\tilde\Pi_{-}$. Left multiplication by the root therefore acts by $+1$ on the right ideal $\tilde\Pi_{+}\mathbb{B}$ and by $-1$ on $\tilde\Pi_{-}\mathbb{B}$, and right multiplication by the root acts by $+1$ on the left ideal $\mathbb{B}\tilde\Pi_{+}$ and by $-1$ on $\mathbb{B}\tilde\Pi_{-}$. Since the pair is complete, every element splits on both sides,

$$ \mathbb{B}=\tilde\Pi_{+}\mathbb{B}\oplus\tilde\Pi_{-}\mathbb{B},\qquad \mathbb{B}=\mathbb{B}\tilde\Pi_{+}\oplus\mathbb{B}\tilde\Pi_{-}, $$

and these are exactly the eigenspace decompositions of the two multiplications; the operator of left multiplication by the root is an involution, its square being the identity because $\tilde\Xi^2=e_t$. The action is collected in the table.

The Standard Idempotents

The algebra carries the standard orthogonal idempotents $\tilde\Pi_{+}=\tfrac12(e_t+i\vec e_z)$ and $\tilde\Pi_{-}=\tfrac12(e_t-i\vec e_z)$, the two idempotents of the root $i\vec e_z$, with $\tilde\Pi_{+}\tilde\Pi_{-}=0$ and $\tilde\Pi_{+} + \tilde\Pi_{-}=e_t$; each is primitive, and they generate the two minimal left ideals.

Put

$$ \tilde\Pi_{+}=\frac{e_t+i\vec e_z}{2},\qquad \tilde\Pi_{-}=\frac{e_t-i\vec e_z}{2}. $$

They are the pair of the root $\tilde\Xi=i\vec e_z$, that is $\tilde\Pi_{+}=\tilde\Pi_{+}(i\vec e_z)$ and $\tilde\Pi_{-}=\tilde\Pi_{-}(i\vec e_z)$ in the notation of §The Two Idempotents of a Root, and $i\vec e_z$ is a root of $+1$, since $(i\vec e_z)^2=i^2\vec e_z^2=(-1)(-1)=e_t$. Hence $\tilde\Pi_{+}^2=\tilde\Pi_{+}$ and $\tilde\Pi_{-}^2=\tilde\Pi_{-}$ by §The Two Idempotents of a Root, and

$$ \tilde\Pi_{+}\tilde\Pi_{-}=\tilde\Pi_{-}\tilde\Pi_{+}=\frac{e_t-(i\vec e_z)^2}{4}=0,\qquad \tilde\Pi_{+} + \tilde\Pi_{-}=e_t . $$

So $\tilde\Pi_{+}$ and $\tilde\Pi_{-}$ are orthogonal idempotents summing to the unit. They are not central, since $\vec e_x$ anticommutes with $i\vec e_z$ and so does not commute with either. Each is primitive, and these two are the standard idempotents of the algebra. The properties of the pair are collected in the table.

Left multiplication by $\vec e_z$ and by $\vec e_y$ acts on them as

$$ \vec e_z\tilde\Pi_{+}=-i\tilde\Pi_{+},\qquad \vec e_z\tilde\Pi_{-}=i\tilde\Pi_{-},\qquad \vec e_y\tilde\Pi_{+}=i\vec e_x\tilde\Pi_{+}, $$

the first because $\vec e_z\tilde\Pi_{+}=(\vec e_z+i\vec e_z^2)/2=(\vec e_z-i)/2=-i\tilde\Pi_{+}$, and similarly for $\tilde\Pi_{-}$. The relations reduce the four products $e_\mu\tilde\Pi_{+}$ to $\tilde\Pi_{+}$ and $\vec e_x\tilde\Pi_{+}$, so that $\{\tilde\Pi_{+},\vec e_x\tilde\Pi_{+}\}$ is a $\mathbb{C}$-basis of the minimal left ideal $\mathbb{B}\tilde\Pi_{+}$. The two ideals, their direct-sum decomposition of $\mathbb{B}$, their bases and their module structure are Ideals and Peirce Decompositions of the Biquaternion Algebra.

The Classification of the Idempotents

Every idempotent of $\mathbb{B}$ is either trivial ($0$ or $e_t$) or of the form $\tilde\Pi=\tfrac12(e_t+\tilde\Xi)$ with $\tilde\Xi^2=e_t$; there are no others.

Theorem (the idempotents). Every idempotent of $\mathbb{B}$ is either trivial ($0$ or $e_t$) or of the form

$$ \boxed{\; \tilde\Pi=\tfrac12 e_t+\tfrac12\tilde\Xi \;}, $$

where $\tilde\Xi\in\mathbb{B}$ is a root of $+1$, that is $\tilde\Xi^2=e_t$. There are no other idempotents in $\mathbb{B}$.

Proof. Write $\tilde\Pi=\Pi_te_t+\vec\Pi$ with $\Pi_t\in\mathbb{C}$ and $\vec\Pi$ pure. Then

$$ \tilde\Pi^2=\bigl(\Pi_t^2-(\vec\Pi,\vec\Pi)\bigr)e_t+2\Pi_t\vec\Pi, $$

and equating this to $\tilde\Pi=\Pi_te_t+\vec\Pi$ gives the two equations

$$ \Pi_t^2-(\vec\Pi,\vec\Pi)=\Pi_t,\qquad 2\Pi_t\vec\Pi=\vec\Pi . $$

If $\vec\Pi=0$ then $\Pi_t^2=\Pi_t$, so $\Pi_t=0$ or $\Pi_t=1$, giving the trivial idempotents. If $\vec\Pi\neq0$ then the second equation gives $\Pi_t=1/2$, and substituting into the first gives

$$ (\vec\Pi,\vec\Pi)=-\tfrac14 . $$

Put $\tilde\Xi=2\vec\Pi$. Then $\tilde\Xi$ is pure and

$$ (\tilde\Xi,\tilde\Xi)=\sum_k(2\Pi_k)^2=4(\vec\Pi,\vec\Pi)=-1, $$

so $\tilde\Xi^2=-(\tilde\Xi,\tilde\Xi)e_t=e_t$: the element $\tilde\Xi$ is a root of $+1$, and $\vec\Pi=\tilde\Xi/2$, whence $\tilde\Pi=\tfrac12 e_t+\tfrac12\tilde\Xi$. The other member of the pair is obtained by replacing $\tilde\Xi$ by $-\tilde\Xi$, which is again a root of $+1$.

The trivial idempotents correspond to the trivial roots $\tilde\Xi=\pm e_t$: the two elements $\tfrac12(e_t+\tilde\Xi)$ are then $e_t$ and $0$.

The Correspondence with the Roots of One

The halving map $H(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi)$ is a bijection from the roots of $+1$ onto the idempotents; the pair map $\tilde\Xi\mapsto(\tilde\Pi_{+}(\tilde\Xi),\tilde\Pi_{-}(\tilde\Xi))$ is a bijection from the roots of $+1$ onto the ordered complete orthogonal pairs of idempotents; and the class $\{\tilde\Xi,-\tilde\Xi\}$ gives the same unordered pair.

Theorem (the halving map). The map

$$ \boxed{\; H(\tilde\Xi)=\tilde\Pi_{+}(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi) \;} $$

is a bijection from the set of roots of $+1$ onto the set of idempotents of $\mathbb{B}$.

Proof. Well defined: if $\tilde\Xi^2=e_t$, then $\tilde\Pi_{+}(\tilde\Xi)^2=\tilde\Pi_{+}(\tilde\Xi)$ by §The Two Idempotents of a Root. Injective: $\tilde\Pi_{+}(\tilde\Xi)=\tilde\Pi_{+}(\tilde\Xi')$ gives $\tilde\Xi=\tilde\Xi'$, since $\tilde\Xi=2\tilde\Pi_{+}(\tilde\Xi)-e_t$. Surjective: the classification theorem says every idempotent is $\tfrac12(e_t+\tilde\Xi)$ or $\tfrac12(e_t-\tilde\Xi)$ for some root $\tilde\Xi$ of $+1$; the first is $\tilde\Pi_{+}(\tilde\Xi)$, and the second is $\tilde\Pi_{-}(\tilde\Xi)=\tilde\Pi_{+}(-\tilde\Xi)$, while $-\tilde\Xi$ is again a root of $+1$.

Theorem (the pair map). The map

$$ \tilde\Xi\longmapsto\bigl(\tilde\Pi_{+}(\tilde\Xi),\tilde\Pi_{-}(\tilde\Xi)\bigr) $$

is a bijection from the set of roots of $+1$ onto the set of ordered pairs $(\tilde\Pi_{+},\tilde\Pi_{-})$ of idempotents that are complete and orthogonal, $\tilde\Pi_{+} + \tilde\Pi_{-}=e_t$ and $\tilde\Pi_{+}\tilde\Pi_{-}=\tilde\Pi_{-}\tilde\Pi_{+}=0$.

Proof. Well defined: the pair of a root is complete and orthogonal by §The Arithmetic of the Pair. Injective: the root is recovered from the pair, $\tilde\Xi=\tilde\Pi_{+} - \tilde\Pi_{-}$, so two roots with the same pair are equal. Surjective: an ordered complete orthogonal pair $(\tilde\Pi_{+},\tilde\Pi_{-})$ gives the element $\tilde\Xi=\tilde\Pi_{+} - \tilde\Pi_{-}$, and

$$ \tilde\Xi^2=\tilde\Pi_{+}^2-\tilde\Pi_{+}\tilde\Pi_{-} - \tilde\Pi_{-}\tilde\Pi_{+} + \tilde\Pi_{-}^2=\tilde\Pi_{+} + \tilde\Pi_{-}=e_t, $$

so $\tilde\Xi$ is a root of $+1$ and the pair is its pair.

The pair map is a bijection, so the idempotent classification and the root classification are the same classification read through it. The three root families and the idempotents they carry correspond as follows.

The signs in a row are correlated, and the third and fourth columns are the two members of the ordered pair.

Consequently the complementary pairs $\{\tilde\Pi,e_t-\tilde\Pi\}$ of idempotents, unordered, are in bijection with the roots of $+1$ modulo the sign identification $\tilde\Xi\sim-\tilde\Xi$, since

$$ \tilde\Pi_{-}(\tilde\Xi)=\tfrac12(e_t-\tilde\Xi)=e_t-\tilde\Pi_{+}(\tilde\Xi)=\tilde\Pi_{+}(-\tilde\Xi). $$

The inverse of the halving map is $H^{-1}(\tilde\Pi)=2\tilde\Pi-e_t$, and the root is the difference of its pair, $\tilde\Xi=\tilde\Pi_{+} - \tilde\Pi_{-}$.

The Families of Idempotents

The idempotents fall into three families: the trivial idempotents $0,e_t$, the non-trivial Hermitian idempotents, and the non-Hermitian idempotents.

The idempotents fall into the three following families.

Since $i\vec u$ is Hermitian for a real unit vector $\vec u$ with $\vec u^2=-e_t$, these are the non-trivial Hermitian idempotents, and they lie in the Hermitian subspace $\mathbb{M}_+$. They are the spectral idempotents of the spectral theorem for a Hermitian element, treated in The Biquaternion Eigenvalue Decomposition. A Hermitian idempotent is self-adjoint, orthogonality being read from the Hermitian form of The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra. For the non-Hermitian root the vector part $\tfrac12(\vec\xi+i\vec\xi')$ carries both a nonzero real part $\vec\xi$ and a nonzero imaginary part $\vec\xi'$, so the idempotent lies in none of the four four-dimensional subspaces $\mathbb{H}_{\mathbb{B}}$, $i\mathbb{H}_{\mathbb{B}}$, $\mathbb{M}_+$ and $\mathbb{M}_-$. The halving map is a bijection, so the trivial roots give the two isolated trivial idempotents, the purely imaginary roots give the two-real-parameter family of non-trivial Hermitian idempotents, and the non-Hermitian roots give the four-real-parameter family of non-Hermitian idempotents. Every non-trivial idempotent is a zero divisor, as §Idempotents and the Zero Divisors shows, so the non-trivial idempotents lie in the zero-divisor set of Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors, while the trivial idempotents $0$ and $e_t$ are outside it, $e_t$ being a unit.

Idempotents and Their Projectors

Every idempotent is the member attached to $+1$ of the pair of a root, and its complement the member attached to $-1$; the standard pair carries the two projections that resolve the identity, and a Hermitian idempotent is self-adjoint for the Hermitian form of The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra and lies in $\mathbb{M}_+$.

Every idempotent is the member attached to $+1$ of the pair of a root: for $\tilde\Pi$ idempotent the element $\tilde\Xi=2\tilde\Pi-e_t$ is a root of $+1$, and $\tilde\Pi_{+}(\tilde\Xi)=\tilde\Pi$ while $\tilde\Pi_{-}(\tilde\Xi)=e_t-\tilde\Pi$. The complement $e_t-\tilde\Pi$ is therefore the member attached to $-1$ of the same pair, an idempotent orthogonal to $\tilde\Pi$ by §The Arithmetic of the Pair, and the direct sum decomposition of the underlying $\mathbb{C}$-module,

$$ \mathbb{B}=\tilde\Pi\mathbb{B}\oplus(e_t-\tilde\Pi)\mathbb{B},\qquad \mathbb{B}=\mathbb{B}\tilde\Pi\oplus\mathbb{B}(e_t-\tilde\Pi), $$

is the one read there at a single root. This is the algebraic content of the statement that idempotents correspond to projectors under multiplication: multiplication by the idempotent is the projector, its complement gives the complementary projector, and the algebra splits into the image of one and the image of the other.

For the standard pair, left multiplication gives the two projections

$$ P_{+}:\tilde Q\longmapsto\tilde\Pi_{+}\tilde Q,\qquad P_{-}:\tilde Q\longmapsto\tilde\Pi_{-}\tilde Q, $$

with

$$ P_{+}^{2}=P_{+},\qquad P_{-}^{2}=P_{-},\qquad P_{+}P_{-}=P_{-}P_{+}=0,\qquad P_{+}+P_{-}=\mathrm{id}. $$

The identity $P_{+}+P_{-}=\mathrm{id}$ is the resolution of the identity; the images are the two minimal right ideals $\tilde\Pi_{+}\mathbb{B}$ and $\tilde\Pi_{-}\mathbb{B}$. Right multiplication gives the two projections onto the minimal left ideals $\mathbb{B}\tilde\Pi_{+}$ and $\mathbb{B}\tilde\Pi_{-}$, with the same identities. The two left projectors are collected in the table.

A Hermitian idempotent, $\tilde\Pi^{*}=\tilde\Pi$, is self-adjoint with respect to the Hermitian form of The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra, which this article names and does not use; it is the kind occurring in the spectral decomposition of a Hermitian element. Since $\tilde\Pi^{*}=\overline{\tilde\Pi^{\natural}}$, the non-trivial Hermitian idempotents are the members of the second family of §The Families of Idempotents, and they lie in $\mathbb{M}_+$.

Idempotents and the Zero Divisors

Every non-trivial idempotent is a zero divisor, and conversely every non-pure zero divisor is the complex multiple $\tilde Q=2Q_t\tilde\Pi$ of a non-trivial idempotent.

A non-trivial idempotent is a zero divisor, since

$$ \tilde\Pi(e_t-\tilde\Pi)=\tilde\Pi-\tilde\Pi^2=0, $$

and both factors are nonzero unless $\tilde\Pi$ is $0$ or $e_t$. Conversely, every non-pure zero divisor — every zero divisor with $Q_t\neq0$ — is a complex multiple of a non-trivial idempotent,

$$ \tilde Q=2Q_t\tilde\Pi,\qquad \tilde\Pi=\frac{\tilde Q}{2Q_t}. $$

The two kinds of zero divisor are collected in the table.

The derivation, from the square relation $\tilde Q^2=2Q_t\tilde Q$ that the non-pure family satisfies, together with the structure of the two families of the zero-divisor set and their distribution among the remarkable subspaces, is Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors.

Idempotents and the Minimal Ideals

The standard idempotents generate the two minimal left ideals, and the algebra is their direct sum.

The principal left ideals $\mathbb{B}\tilde\Pi_{+}$ and $\mathbb{B}\tilde\Pi_{-}$ are the two minimal left ideals of the standard pair, and the completeness and orthogonality of the pair give the direct sum

$$ \mathbb{B}=\mathbb{B}\tilde\Pi_{+}\oplus\mathbb{B}\tilde\Pi_{-}. $$

The two ideals are collected in the table.

The minimality is the primitive character of the pair: the only idempotents of $\tilde\Pi_{+}\mathbb{B}\tilde\Pi_{+}$ and $\tilde\Pi_{-}\mathbb{B}\tilde\Pi_{-}$ are $0$ and the identity of the corner, so neither ideal contains a proper nonzero ideal of its own. The Peirce decomposition, the corner algebras and the matrix units of the two ideals are Ideals and Peirce Decompositions of the Biquaternion Algebra.

Idempotents and the Involutions

For the standard pair, the natural conjugation and the bar exchange $\tilde\Pi_{+}$ and $\tilde\Pi_{-}$, while the star fixes both.

The action of the involutions on the idempotents is collected in the table.

The natural conjugation changes the sign of the vector part and fixes the central $i$, so it sends $i\vec e_z$ to $-i\vec e_z$ and exchanges the two idempotents. The bar conjugates the complex coefficient $i$ and fixes $\vec e_z$, so it also exchanges them. The star is the composite $\bar{\cdot}\circ{}^{\natural}$, so it fixes each: $\tilde\Pi_{+}^{*}=\tilde\Pi_{+}$ and $\tilde\Pi_{-}^{*}=\tilde\Pi_{-}$. The trivial idempotents $0$ and $e_t$ are fixed by all three. The non-trivial fixed idempotents of the star are the non-trivial Hermitian idempotents of §The Families of Idempotents, and the trivial idempotents $0$ and $e_t$ are fixed as well; the natural conjugation and the bar exchange the members of the standard pair without preserving the individual idempotent.

Summary

The roots of $+1$ in $\mathbb{B}$ are, by the vector–scalar split and the product formula $\tilde\Xi^2=(\Xi_t^2-(\vec\Xi,\vec\Xi))e_t+2\Xi_t\vec\Xi$:

1. Trivial roots: $\tilde\Xi=\pm e_t$.
2. Purely imaginary roots: $\tilde\Xi=\pm i\vec u$, where $\vec u$ is a unit real vector with $\vec u^2=-e_t$, a two-real-parameter family.
3. Non-Hermitian roots: $\tilde\Xi=\vec\xi+i\vec\xi'$ with $\vec\xi\neq0$ and the two real conditions $\xi_x^2+\xi_y^2+\xi_z^2-(\xi'_x)^2-(\xi'_y)^2-(\xi'_z)^2=-1$ and $\xi_x\xi'_x+\xi_y\xi'_y+\xi_z\xi'_z=0$, a four-real-parameter family.

All roots except the trivial ones are pure. Every root of $+1$ is a unit, with $\tilde\Xi^{-1}=\tilde\Xi$. The roots of $-1$ are read off the roots of $+1$ by the central imaginary, $\tilde\Xi\mapsto i\tilde\Xi$, a bijection that turns the three families into the two central scalars $\pm i$, the family of pure real roots $\pm\vec u$, and the family $-\vec\xi'+i\vec\xi$; the mechanism needs a central scalar of square $-1$ and is not universal.

The idempotents are the trivial elements $0$ and $e_t$ and the elements $\tilde\Pi=\tfrac12(e_t\pm\tilde\Xi)$ with $\tilde\Xi$ a root of $+1$. Each root carries the two idempotents $\tilde\Pi_{+}(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi)$ and $\tilde\Pi_{-}(\tilde\Xi)=\tfrac12(e_t-\tilde\Xi)=e_t-\tilde\Pi_{+}(\tilde\Xi)$, a complete orthogonal pair: their sum is $e_t$, their product is $0$, their difference is the root, and multiplication by the root acts by $+1$ on $\tilde\Pi_{+}$ and $-1$ on $\tilde\Pi_{-}$, so that the pair splits the algebra into the eigenspaces of that multiplication. The halving map $H(\tilde\Xi)=\tfrac12(e_t+\tilde\Xi)$ is a bijection from the roots of $+1$ onto the idempotents, with inverse $H^{-1}(\tilde\Pi)=2\tilde\Pi-e_t$, and the class $\{\tilde\Xi,-\tilde\Xi\}$ gives the complementary pair $\{\tilde\Pi_{+},\tilde\Pi_{-}\}$. The three families of roots give the trivial idempotents, the non-trivial Hermitian idempotents in $\mathbb{M}_+$ (a two-real-parameter family), and the non-Hermitian idempotents, lying in none of the four four-dimensional subspaces (a four-real-parameter family). The standard pair $\tilde\Pi_{+}=\tfrac12(e_t+i\vec e_z)$ and $\tilde\Pi_{-}=\tfrac12(e_t-i\vec e_z)$ is orthogonal, sums to $e_t$, is primitive, and generates the two minimal left ideals. An idempotent and its complement split the algebra, $\mathbb{B}=\tilde\Pi\mathbb{B}\oplus(e_t-\tilde\Pi)\mathbb{B}$, the algebraic form of a projector and its complementary projector. Every non-trivial idempotent is a zero divisor, and every non-pure zero divisor is the complex multiple $2Q_t\tilde\Pi$ of a non-trivial idempotent, developed in Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors.

Summary of Notation

Further Reading

• Definitions for the Study of the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the general definition of a root of a central value, of a square and of an idempotent, with the orthogonal, complete and primitive families, the Hermitian idempotent and the Peirce decomposition.
• Square Roots of a General Element in the Biquaternion Algebra, for the roots of the general quadratic $\tilde\Xi^2=\tilde Q$, of which the roots of $+1$ here are one case.
• Square Roots of Zero and the Nilpotents of the Biquaternion Algebra, for the roots of $0$ and the nilpotents, the second central value.
• Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, for the two families of the zero-divisor set, of which the non-pure family consists of the complex multiples of the non-trivial idempotents.
• Ideals and Peirce Decompositions of the Biquaternion Algebra, for the minimal left ideals the standard idempotents generate and the Peirce decomposition they drive.
• Square Roots of One and the Idempotents in the Algebraic Structures over the Biquaternion $\mathbb{C}$ Space, for the four root sets and the four idempotent sets compared.
• Square Roots of One and the Idempotents of the Split-Complex Algebra, the real commutative case, where the roots of $+1$ and the idempotents are four points and there is no root of $-1$.
• Square Roots of One and the Idempotents of the Algebra of the Reduced Biquaternions, the complex commutative case, where the roots of $+1$ and the idempotents are four points and the roots of $-1$ are four.
• S. J. Sangwine, "Biquaternion (complexified quaternion) roots of $-1$", Advances in Applied Clifford Algebras 16 (2006) 63–68.
• S. J. Sangwine and D. Alfsmann, "Determination of the biquaternion divisors of zero, including the idempotents and nilpotents", Advances in Applied Clifford Algebras 20 (2010) 401–416.
• William Kingdon Clifford, "Preliminary Sketch of Biquaternions" (1873), for the first systematic treatment of biquaternions.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for idempotents and minimal left ideals in Clifford algebras.
• J. P. Ward, Quaternions and Cayley Numbers (Kluwer, 1997), for the idempotents of the biquaternion algebra.