The Biquaternion Involution Lattice: Hermitian, Anti-Hermitian and Reversal

Introduction

The biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$ carries four distinguished conjugations, and almost every operation in the relativistic articles is one of them. The biquaternion norm is built from the first, the reality of an observable is decided by the second, the adjoint of an operator is the third, and the algebra's real structure is the fourth. This article fixes the four, derives their composition law, identifies the real subspaces they single out, and states the two pairings and the adjoint identities that the operator article uses.

The four maps are quaternion conjugation (also called reversal or conjugation by the quaternion conjugate) ${}^{\natural}$, complex conjugation $\bar{\cdot}$, Hermitian conjugation ${}^{*} = ({}^{\natural})^{\,*}$, and anti-Hermitian conjugation $\flat = -{}^{*}$. Their names are inherited unchanged from the companion Conventions in the Biquaternion Universe, and nothing here renames them. Three facts organise the article.

First, the composition law. Quaternion conjugation and complex conjugation commute, and together with the identity they close into the Klein four-group: ${}^{*} = {}^{\natural}\circ\bar{\cdot}$, and every element is its own inverse. The anti-Hermitian conjugation is the Hermitian one twisted by the central element $-1$, so $\flat = -{}^{*}$ is a fifth involution but not a fifth member of that group; it obeys an order-reversing law with a sign, $(\tilde{A}\tilde{B})^\flat = -\tilde{B}^\flat\tilde{A}^\flat$. The group generated by the four conjugations together with the central scalars $\{\pm1,\pm i\}$ has order sixteen and was checked to have exactly sixteen distinct elements, the central factor contributing a $\mathbb{Z}_4$ and the conjugations a $V_4$.

Second, the lattice. Each involution has a fixed subspace, and the four fixed subspaces are four of the framework's six distinguished subspaces: $$ \mathbb{C}_{\mathbb{B}} = \{\tilde{Q} : \tilde{Q}^{\natural} = \tilde{Q}\} \ (\dim 2), \qquad \mathbb{H}_{\mathbb{B}} = \{\tilde{Q} : \tilde{Q}^* = \tilde{Q}\} \ (\dim 4, \text{ a subalgebra}), $$ $$ \mathbb{M}_{+} = \{\tilde{Q} : \tilde{Q}^{*} = \tilde{Q}\} \ (\dim 4), \qquad \mathbb{M}_{-} = \{\tilde{Q} : \tilde{Q}^\flat = \tilde{Q}\} \ (\dim 4). $$ The remaining two are the anti-fixed spaces: the vector subspace $\mathrm{Vect}(\mathbb{B})$, the $-1$ eigenspace of ${}^{\natural}$, and the antiquaternion subspace $i\mathbb{H}_{\mathbb{B}}$, the $-1$ eigenspace of $\bar{\cdot}$. Named one by one in the order used throughout the framework, the six are the center subspace $\mathbb{C}_{\mathbb{B}}$, the vector subspace $\mathrm{Vect}(\mathbb{B})$, the quaternion subspace $\mathbb{H}_{\mathbb{B}}$, the antiquaternion subspace $i\mathbb{H}_{\mathbb{B}}$, the informational sector $\mathbb{M}_+$ and the material sector $\mathbb{M}_-$; the table below lists all six with their involution, dimension and basis, and the lattice developed in this article is the lattice of the four fixed spaces among them, with the inclusions, the dimensions and the intersections tabulated and verified below. The two four-dimensional forms $\mathbb{M}_+$ and $\mathbb{M}_-$ are the eigenspaces of ${}^{*}$ with eigenvalues $+1$ and $-1$, so they are complementary, $\mathbb{B} = \mathbb{M}_+\oplus\mathbb{M}_-$; the companion articles call them the informational and material sectors. All four conjugations preserve each sector; the central multiplications $m_{\pm i}$ exchange them, and so does every product of one of those with a conjugation — eight of the sixteen operations in all — and this is the sharp sense in which the conjugations and the complex structure are different operations.

Third, the adjoints. The reason the lattice is not merely bookkeeping is that adjoints of operators are read off it. Left multiplication by $\tilde{A}$ has adjoint left multiplication by $\tilde{A}^{*}$, and the gradient of the operator article has adjoint $-\tilde{\nabla}^{\natural}$; the two signs cancel in the d'Alembertian, which is self-adjoint. The reversal is the anti-automorphism that builds the biquaternion norm, and the fixed space of $\flat$ is the real form in which the first-order kernels live. The operator article uses these facts; this article derives them.

The article is organised as follows. The four conjugations are defined and typed, the Klein four-group is established with its composition table, the fixed spaces and their lattice are computed. The two pairings — the bilinear one built from reversal and the Hermitian one built from ${}^{*}$ — are treated next, with their signatures and invariance, and the article closes with the adjoint identities and the central complex structure.

• Companion article Conventions in the Biquaternion Universe, for the four conjugations and the two sectors as the series fixes them.
• Companion article Relations Between Subspaces, for the six distinguished subspaces named one by one, their three decompositions, and the six-subspace intersection table restricted here to the four fixed spaces.
• Companion article The Anti-Hermitian Subspace M- as the Material Sector, for the material sector and its four-vector reading.
• Companion article The Hermitian Subspace M+ as the Informational Sector, for the Hermitian sector and the observables.
• Companion article The Dirac Algebra and Biquaternions — A Dictionary, for reversal as Clifford reversion.
• Companion article Antilinear Structure and the Two Kinds of Mass in Biquaternionic Form, for $\flat$ as the real structure and the antilinear operations built from it.
• Companion article Exercise: Chirality and the Weyl Spinors, for the conjugate spinor structures and the real-time writing of the d'Alembertian.

The Four Conjugations

The algebra has the $\mathbb{C}$-basis $e_0 = 1, e_1, e_2, e_3$ with $e_1^2 = e_2^2 = e_3^2 = -e_0$, $e_1e_2 = e_3$, and $e_je_k = -e_ke_j$ for $j\neq k$; the scalar imaginary $i$ satisfies $i^2 = -1$ and commutes with every $e_\mu$. The four maps are defined on the basis and extended by the rules of type stated with each.

Quaternion conjugation (reversal). The map ${}^{\natural}$ fixes the scalar unit and negates the vector units, $$ \bar{e}_0 = e_0 , \qquad \bar{e}_k = -e_k , \qquad \bar{i} = i , $$ and is extended as a $\mathbb{C}$-linear anti-automorphism: $$ (\tilde{A}\tilde{B})^{\natural} = \tilde{B}^{\natural}\,\tilde{A}^{\natural} , \qquad \overline{\alpha\tilde{A} + \beta\tilde{B}} = \alpha\tilde{A}^{\natural} + \beta\tilde{B}^{\natural} , \qquad \alpha,\beta \in \mathbb{C}. $$ It reverses the order of a product and fixes the complex scalars, which is exactly the Clifford reversal of the even algebra; the corpus uses the name "reversal" for it. Its fixed space is $\mathbb{C}_{\mathbb{B}}$.

Complex conjugation. The map $\bar{\cdot}$ conjugates the scalar imaginary and fixes the quaternion units, $$ i^* = -i , \qquad e_\mu^* = e_\mu , $$ and is extended as a $\mathbb{C}$-antilinear automorphism: $$ (\tilde{A}\tilde{B})^* = \tilde{A}^*\tilde{B}^* , \qquad (\alpha\tilde{A} + \beta\tilde{B})^* = \alpha^*\tilde{A}^* + \beta^*\tilde{B}^* . $$ It preserves the order of a product and conjugates the scalars, and its fixed space is the real-quaternion algebra $\mathbb{H}_{\mathbb{B}}$.

Hermitian conjugation. The map ${}^{*}$ is the composition $$ \tilde{Q}^{*} := \bigl(\bar{\tilde{Q}}\bigr)^{\natural} = \bigl(\tilde{Q}^{\natural}\bigr)^{*} , $$ so it negates both the vector units and the scalar imaginary. It is a $\mathbb{C}$-antilinear anti-automorphism, $$ (\tilde{A}\tilde{B})^{*} = \tilde{B}^{*}\tilde{A}^{*} , \qquad (\alpha\tilde{A})^{*} = \alpha^*\tilde{A}^{*} , $$ Its fixed space is $\mathbb{M}_+$, the informational sector.

Anti-Hermitian conjugation. The map $\flat$ is the Hermitian conjugation with a sign, $$ \tilde{Q}^\flat := -\,\tilde{Q}^{*} . $$ It is again an involution — applying it twice returns the element — but it is an anti-automorphism only up to the central sign, $$ (\tilde{A}\tilde{B})^\flat = -\,\tilde{B}^\flat\tilde{A}^\flat . $$ Its fixed space is $\mathbb{M}_-$, the material sector, and the companion article records $\flat$ as the algebra's real structure.

The action of the four maps on the eight real basis elements $e_0,e_1,e_2,e_3,ie_0,ie_1,ie_2,ie_3$ is a signed permutation, and the sign table is worth displaying because every later identity is read from it:

The table was computed by applying each map to each basis element and reading off the signed representative; each map was verified to be an involution on all eight, and the table is the exact one.

Which of the four maps are words in the products. Reversal is built from the algebra's own operations. With the quaternion units, $$ \tilde{Q}^{\natural} = -\tfrac12\bigl(\tilde{Q} + e_1\tilde{Q}e_1 + e_2\tilde{Q}e_2 + e_3\tilde{Q}e_3\bigr) , \qquad e_1\tilde{Q}e_1 + e_2\tilde{Q}e_2 + e_3\tilde{Q}e_3 = -3S(\tilde{Q}) + V(\tilde{Q}) , $$ verified on three hundred random complex biquaternions; the four-term combination is therefore the difference $S - V$ on the fixed space $\mathbb{C}_{\mathbb{B}}$. This is the statement, due to Sangwine, Ell and Le Bihan, that the quaternion conjugate may be expressed "in terms of multiplications and additions": the formula uses the product, the sum and constants drawn from the algebra, and no division. Complex conjugation is not of this kind, and no formula of this kind exists for it. A word in $\tilde{Q}$ with constant coefficients built from $+$ and $\cdot$ cannot compute $\bar{\cdot}$. Suppose a polynomial function $P$ with coefficients in $\mathbb{B}$ satisfied $P(\tilde{Q}) = \tilde{Q}^*$ for all $\tilde{Q}$. On an element $z$ of the centre the coefficients commute with $z$, so $P(z) = \sum_k a_kz^k$; at a real $z$ this must equal $z$, and a polynomial function with coefficients in the finite-dimensional real space $\mathbb{B}$ that vanishes on infinitely many points is the zero polynomial, so $P$ is the identity word, and at $z = i$ it gives $i$, whereas $i^* = -i$. The same argument excludes ${}^{*}$ and $\flat$, whose action on the centre is also conjugation. So reversal is the only one of the four maps that is a word in the products, and ${}^{*} = {}^{\natural}\circ\bar{\cdot}$ composes a derived map with a primitive one. Sangwine, Ell and Le Bihan state the contrast explicitly: the involution-based formulas "require multiplication and addition only", whereas the formulas whose four terms carry the four conjugates "contain complex conjugates, which are primitive operations that cannot be expressed in terms of multiplications and additions". The consequence carried through the article is that a formula decomposing a biquaternion into its complex scalar and vector parts is available in the products, while any formula that separates the real parts from the imaginary parts needs the primitive $\bar{\cdot}$.

The Composition Law

The Klein four-group

Quaternion conjugation is $\mathbb{C}$-linear and complex conjugation is $\mathbb{C}$-antilinear; acting on independent data — the basis $e_k$ and the scalar $i$ — they commute: $$ \bigl(\bar{\tilde{Q}}\bigr)^{\natural} = \bigl(\tilde{Q}^{\natural}\bigr)^{*} \qquad\text{for all } \tilde{Q} \in \mathbb{B} . $$ It follows that $\{\mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*}\}$ is closed, and since each element is its own inverse and the group is abelian it is the Klein four-group $V_4 = \mathbb{Z}_2\times\mathbb{Z}_2$. The full composition table, computed on five random complex biquaternions, is

so that ${}^{*} = {}^{\natural}\circ\bar{\cdot} = \bar{\cdot}\circ{}^{\natural}$, and the three non-identity elements are the three ways of negating a subset of the signs $\{e_k\text{-sign},\,i\text{-sign}\}$: reversal negates the $e_k$ sign, complex conjugation the $i$ sign, and Hermitian conjugation both. Every entry was verified as an identity of maps on random elements, with no failure.

The anti-Hermitian involution and the sign twist

The map $\flat = -{}^{*}$ is an involution, $$ (\tilde{Q}^\flat)^\flat = \bigl(-\tilde{Q}^{*}\bigr)^\flat = -\bigl(-\tilde{Q}^{*}\bigr)^{*} = (\tilde{Q}^{*})^{*} = \tilde{Q} , $$ using the reality of $-1$ and the antilinearity of ${}^{*}$. It is not in $V_4$, because $V_4$ consists of the four maps that are either linear or antilinear automorphisms or anti-automorphisms with no scalar twist, and $\flat$ carries the extra central factor. Composing it with reversal and with complex conjugation gives the twisted versions of the table, $$ \flat\circ{}^{\natural} = -\,\bar{\cdot} , \qquad \flat\circ\bar{\cdot} = -\,{}^{\natural} , \qquad \flat\circ{}^{*} = \mathrm{id}\cdot(-1) , $$ where the last means $\flat\circ{}^{*} = -\,\mathrm{id}$ as a map, again verified directly. The twist is a central sign and therefore commutes with everything; this is why $\flat$ can be used as a conjugation at all, and why its fixed space is nevertheless a genuine real form.

The group generated by the conjugations and the central scalars

Adjoining the central multiplications $m_\lambda : \tilde{Q}\mapsto\lambda\tilde{Q}$ for $\lambda \in \{1,i,-1,-i\}$ to the four conjugations produces a group of order sixteen, $$ G = \langle m_i,\, {}^{\natural},\, \bar{\cdot}\rangle , \qquad |G| = 16 . $$ The group is not a direct product, and the reason is a sign that is easy to miss. Reversal commutes with $m_i$, because it is $\mathbb{C}$-linear and $i$ is central, $$ {}^{\natural}\circ m_i = m_i\circ{}^{\natural} , $$ but complex conjugation anticommutes with it, because it conjugates the scalar: $$ \bar{\cdot}\circ m_\lambda = m_{\bar\lambda}\circ\bar{\cdot} , \qquad\text{so in particular}\qquad \bar{\cdot}\circ m_i = m_{-i}\circ\bar{\cdot} . $$ Consequently $G$ is non-abelian, and it is the semidirect product $$ G \cong \mathbb{Z}_4 \rtimes V_4 \cong D_8 \times \mathbb{Z}_2 , $$ where $D_8 = \langle m_i, \bar{\cdot}\rangle$ is the dihedral group of order eight — $m_i$ a four-fold rotation and $\bar{\cdot}$ a reflection that inverts it — and the commuting reversal generates the central $\mathbb{Z}_2$. Reading $G$ as $\mathbb{Z}_4\times V_4$ would make it abelian with a centre of order sixteen; the computed centre has order four, $\{e_0,\, m_{-1},\, {}^{\natural},\, m_{-1}{}^{\natural}\}$, and the abelian reading is thereby excluded. The sixteen composite maps were enumerated explicitly: they are pairwise distinct, their element orders are $1$ for the identity, $2$ for eleven elements and $4$ for four, and the four of order four are $m_{\pm i}$ and $m_{\pm i}\circ{}^{\natural}$. Composing $m_{\pm i}$ with $\bar{\cdot}$ or with ${}^{*}$ gives an involution instead, because those conjugations invert $m_i$; it is only reversal, which commutes with it, that preserves the order. The structural point survives in corrected form: the conjugations form a four-element group and the complex structure a four-element group, but they do not commute, and it is their semidirect product — not either factor alone, and not a direct product — that describes the algebra's full group of signs.

A biquaternion need not commute with its complex conjugate

An element commutes with its reversal in either order, and that is the identity behind the biquaternion norm, $$ \tilde{Q}\tilde{Q}^{\natural} = \tilde{Q}^{\natural}\tilde{Q} = N(\tilde{Q}) . $$ It need not commute with its complex conjugate. Writing $\tilde{Q} = q_r + iq_i$ with $q_r$ and $q_i$ real quaternions, the two products are $$ \tilde{Q}\tilde{Q}^* = q_r^2 + q_i^2 + i\,(q_iq_r - q_rq_i) , \qquad \tilde{Q}^*\tilde{Q} = q_r^2 + q_i^2 - i\,(q_iq_r - q_rq_i) , $$ verified on four hundred random complex biquaternions. They are a complex-conjugate pair, and they differ by twice the commutator of the two halves, carried by the central unit: the failure of $\tilde{Q}$ and $\tilde{Q}^*$ to commute is exactly the failure of $q_r$ and $q_i$ to commute. The fact is worth recording because the two maps commute as maps, as established above, and because a complex scalar does commute with its complex conjugate; the order of the product of a biquaternion with its complex conjugate is nevertheless not free.

The exceptions are exact, and they are the three ways the commutator of the halves can vanish. - Quaternions, $q_i = 0$: both products reduce to $q_r^2$, and there is nothing to commute with. - Imaginary biquaternions, $q_r = 0$: both products reduce to $q_i^2$. - Co-planar elements, whose two halves lie on a common axis and therefore commute: both products reduce to $q_r^2 + q_i^2$.

A concrete instance is $\tilde{Q} = e_0 + e_1 + ie_2$, with $q_r = e_0+e_1$ and $q_i = e_2$. Its biquaternion norm is $N(\tilde{Q}) = 1$, so it is invertible, and $$ \tilde{Q}\tilde{Q}^* = -e_0 + 2e_1 - 2ie_3 , \qquad \tilde{Q}^*\tilde{Q} = -e_0 + 2e_1 + 2ie_3 , \qquad \tilde{Q}\tilde{Q}^* - \tilde{Q}^*\tilde{Q} = -4ie_3 \neq 0 . $$ The two products are not the biquaternion norm and are not scalars; the biquaternion norm is the reversal product $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural}$, which is a complex scalar by construction. Reading the formula against the sign table, the non-commutation is the one place where the $e_k$ signs and the $i$ sign of two conjugations do not combine into a single sign choice: reversal and complex conjugation act on independent data, yet the products they build need not agree.

The Lattice of Fixed Spaces

The six subspaces and the four fixed spaces

Each involution sorts the algebra into its $+1$ and $-1$ eigenspaces, and the framework's six distinguished subspaces are exactly these. Solving the linear conditions, with dimensions computed by exact row reduction on the real linear system, gives them one by one in the order used throughout:

The four fixed spaces are those of ${}^{\natural}$, $\bar{\cdot}$ and ${}^{*}$, together with $\mathbb{M}_-$, which is the fixed space of $\flat = -{}^{*}$; the two rows marked anti-fixed are the $-1$ eigenspaces of ${}^{\natural}$ and $\bar{\cdot}$, and $\mathbb{M}_-$ is the $-1$ eigenspace of ${}^{*}$. The table is the sign table above read one grade at a time, and every entry was verified on the eight real basis elements.

Which of the six are closed under products, and which under brackets, is tabulated in the companion Relations Between Subspaces; the lattice of this article needs only the four fixed spaces. Two of the six are subalgebras: the center subspace $\mathbb{C}_{\mathbb{B}}$ and the quaternion subspace $\mathbb{H}_{\mathbb{B}}$ are closed under multiplication, while the vector subspace $\mathrm{Vect}(\mathbb{B})$, the antiquaternion subspace $i\mathbb{H}_{\mathbb{B}}$, the informational sector $\mathbb{M}_+$ and the material sector $\mathbb{M}_-$ are not closed. The four fixed-space dimensions were checked as the nullity of $(\text{map} - \mathrm{id})$ in each case.

The two four-dimensional forms are complementary. Since ${}^{*}$ is an involution its eigenvalues are $\pm1$, and $$ \mathbb{M}_+ = \ker({}^{*} - \mathrm{id}) , \qquad \mathbb{M}_- = \ker({}^{*} + \mathrm{id}) , \qquad \mathbb{B} = \mathbb{M}_+ \oplus \mathbb{M}_- , $$ with the projections $\frac12(\mathrm{id}+{}^{*})$ and $\frac12(\mathrm{id}-{}^{*})$. The subspaces are the framework's primary decomposition: the informational sector $\mathbb{M}_+$ is the space of Hermitian elements considered as observables, and the material sector $\mathbb{M}_-$ is the space of anti-Hermitian elements considered as four-vectors. The companion articles The Hermitian Subspace M+ as the Informational Sector and The Anti-Hermitian Subspace M- as the Material Sector develop the interpretation; the algebra of the split is what is established here.

The four graded projections

The four maps of $V_4$ combine into four projections, one for each character of the group: $$ \Pi_S = \tfrac14(\mathrm{id} + {}^{\natural} + \bar{\cdot} + {}^{*}) , \qquad \Pi_B = \tfrac14(\mathrm{id} - {}^{\natural} + \bar{\cdot} - {}^{*}) , $$ $$ \Pi_V = \tfrac14(\mathrm{id} - {}^{\natural} - \bar{\cdot} + {}^{*}) , \qquad \Pi_P = \tfrac14(\mathrm{id} + {}^{\natural} - \bar{\cdot} - {}^{*}) . $$ Applied to a biquaternion they pick out its four real components, $$ \Pi_S\tilde{Q} = \Re S(\tilde{Q}) , \quad \Pi_B\tilde{Q} = \Re V(\tilde{Q}) , \quad \Pi_V\tilde{Q} = i\,\Im V(\tilde{Q}) , \quad \Pi_P\tilde{Q} = i\,\Im S(\tilde{Q}) , $$ each written as a multiple of the basis element it names. Every $\Pi$ is idempotent, any two of them multiply to zero, and the four sum to the identity: $$ \Pi_a^2 = \Pi_a , \qquad \Pi_a\Pi_b = 0 \ \ (a \neq b) , \qquad \Pi_S + \Pi_B + \Pi_V + \Pi_P = \mathrm{id} . $$ The four identities were verified on the basis elements and separately on random elements. They are the four minimal idempotents of the group algebra of $V_4$, one for each character, which is why exactly four such combinations arise and why their images are four real lines of the algebra. Read through the geometric-algebra dictionary of the companion The Dirac Algebra and Biquaternions — A Dictionary, the four components are the four grades: $\Re S$ is the scalar, $\Re V$ the bivector, $i\Im V$ the vector, and $i\Im S$ the pseudoscalar. In this article's language the same four lines span the fixed spaces and the anti-fixed one below them: $\Re S$ and $i\Im S$ span the center subspace $\mathbb{C}_{\mathbb{B}}$, while $\Re V$ and $i\Im V$ span the vector subspace $\mathrm{Vect}(\mathbb{B})$, the elements of zero scalar part. The two-term projections above them are the coarse versions already met — $\frac12(\mathrm{id}\pm{}^{\natural})$ give the complex scalar and vector parts $S$ and $V$, $\frac12(\mathrm{id}\pm\bar{\cdot})$ give the componentwise real and imaginary parts, and $\frac12(\mathrm{id}\pm{}^{*})$ give the two sectors.

Which combinations are algebraic is decided by the previous section: the two-term combinations in ${}^{\natural}$ alone are words in the products, whereas the four-term projections contain $\bar{\cdot}$ and are primitive in the same sense. A worked instance, verified: for $\tilde{Q} = (1+2i)e_0 + (3+4i)e_1 + (5+6i)e_2 + (7+8i)e_3$ the four projections are $e_0$, $3e_1 + 5e_2 + 7e_3$, $i(4e_1 + 6e_2 + 8e_3)$ and $2ie_0$.

The inclusion lattice and the intersections

The fixed spaces are not nested. Their intersections are the interesting part of the lattice, and they were computed exactly. The table below is the restriction to the four fixed spaces of the six-subspace intersection table of the companion Relations Between Subspaces, where the two anti-fixed spaces are also carried; the entries agree with that table on the common rows and columns.

The entries are the real dimensions of the pairwise intersections. Three of them carry the structure. $\mathbb{C}_{\mathbb{B}}$ meets the informational sector in the real scalars, $\mathbb{C}_{\mathbb{B}}\cap\mathbb{M}_+ = \mathbb{R}e_0$, and the material sector in the imaginary scalars, $\mathbb{C}_{\mathbb{B}}\cap\mathbb{M}_- = \mathbb{R}(ie_0)$; hence it is the direct sum of the two sectors' time axes, $$ \mathbb{C}_{\mathbb{B}} = \mathbb{R}e_0 \oplus \mathbb{R}(ie_0) = \bigl(\mathbb{C}_{\mathbb{B}}\cap\mathbb{M}_+\bigr)\oplus\bigl(\mathbb{C}_{\mathbb{B}}\cap\mathbb{M}_-\bigr) . $$ $\mathbb{H}_{\mathbb{B}}$ meets the informational sector in the time axis, $\mathbb{H}_{\mathbb{B}}\cap\mathbb{M}_+ = \mathbb{R}e_0$, and the material sector in the pure-vector space, $\mathbb{H}_{\mathbb{B}}\cap\mathbb{M}_- = \mathbb{R}\{e_1,e_2,e_3\}$; thus the real quaternions split by sector as $\mathbb{H}_{\mathbb{B}} = \mathbb{R}e_0\oplus\mathbb{R}\{e_1,e_2,e_3\}$ as well. The two sectors meet only at zero, $\mathbb{M}_+\cap\mathbb{M}_- = \{0\}$, which is the statement of complementarity. The lattice of fixed spaces is the collection of these four subspaces with the inclusions just listed; it is a four-element antichain in the middle, with the scalar line as the minimal non-trivial element common to the algebra-at-large.

The sectors are preserved by every conjugation

A fact that is easy to expect and false is that the conjugations mix the sectors. They do not. Building a Hermitian element $\tilde{H} = \tilde{R}+\tilde{R}^{*}$ and an anti-Hermitian element $\tilde{Q} = \tilde{R}-\tilde{R}^{*}$ from a random biquaternion $\tilde{R}$, and applying each of the four maps to each, gives $$ \tilde{H}^{\natural},\ \bar{\cdot}\tilde{H},\ {}^{*}\tilde{H},\ \flat\tilde{H} \in \mathbb{M}_+ , \qquad \tilde{Q}^{\natural},\ \bar{\cdot}\tilde{Q},\ {}^{*}\tilde{Q},\ \flat\tilde{Q} \in \mathbb{M}_- , $$ verified on two hundred random cases with no exception. The reason is visible in the sign table: each conjugation either fixes or negates the pair (the $e_k$ sign, the $i$ sign) as a whole, and the condition for membership in a sector is exactly a relation between those two signs, which a simultaneous transformation preserves. What does exchange the sectors is an odd power of the central multiplication $$ i\,\mathbb{M}_+ = \mathbb{M}_- , \qquad i\,\mathbb{M}_- = \mathbb{M}_+ , \qquad N(i\tilde{Q}) = -N(\tilde{Q}) , $$ verified on Hermitian and anti-Hermitian random elements: multiplying by $i$ converts a real scalar and imaginary vector into an imaginary scalar and real vector, and vice versa. The conjugations are a group of four sign choices on the basis; the complex structure is a fourth-root-of-unity action on the coefficients; reversal commutes with $m_i$ but complex conjugation anticommutes with it, they generate the order-sixteen group above, and the sector exchange is the eight-element set $m_{\pm i}V_4$ of operations containing an odd power of $m_i$. This is the algebraic reason the companion articles can speak of a "material" and an "informational" reading of one complex field without one conjugation turning into the other. One product is asymmetric in the two maps, and it is recorded with the composition law above: the reversal product is the scalar $N(\tilde{Q})$ in either order, while the complex-conjugate products $\tilde{Q}\tilde{Q}^*$ and $\tilde{Q}^*\tilde{Q}$ need not agree.

The Two Pairings

The bilinear pairing and the biquaternion norm

Reversal defines a complex-bilinear pairing $$ B(\tilde{A},\tilde{B}) := \mathrm{Sc}\bigl(\tilde{A}^{\natural}\tilde{B}\bigr) , \qquad B(\tilde{A},\tilde{A}) = \mathrm{Sc}\bigl(\tilde{A}^{\natural}\tilde{A}\bigr) = \mathrm{Sc}\,N(\tilde{A}) , $$ which is symmetric, $B(\tilde{A},\tilde{B}) = B(\tilde{B},\tilde{A})$, verified on random pairs, and reduces to the biquaternion norm on the diagonal. Read on the two sectors, its signature is the framework's metric structure:

The verification is the biquaternion-norm computation of the companion Conventions in the Biquaternion Universe: for $\tilde{Q} = i v_0e_0 + \mathbf{v} \in \mathbb{M}_-$ the diagonal is $N(\tilde{Q}) = -v_0^2 + \mathbf{v}^2$, the Minkowski interval, and for a real quaternion it is $\sum_\mu Q_\mu^2$. The lattice therefore carries the two signatures: the bilinear pairing is Lorentzian on the material sector and positive definite on the real quaternions, and it is the same level-one form read on the two bases, not a pair of independent choices. This is the algebraic content of the companion article's three-level metric discussion, and the reason a sign difference between the two sectors is never an inconsistency.

When the pairing vanishes. Sangwine, Ell and Le Bihan give the criterion for $B(\tilde{A},\tilde{B}) = 0$ from the coordinate form of the same pairing, $$ B(\tilde{A},\tilde{B}) = \sum_\mu A_\mu B_\mu , $$ a complex scalar because the vector parts cancel; this is the pairing just computed, not a second one, and its symmetry and bilinearity are those already stated. Splitting both arguments into real and imaginary halves, $\tilde{A} = A_r + iA_i$ and $\tilde{B} = B_r + iB_i$ with real-quaternion halves, $$ B(\tilde{A},\tilde{B}) = B(A_r,B_r) - B(A_i,B_i) + i\bigl(B(A_r,B_i) + B(A_i,B_r)\bigr) , $$ so orthogonality means that a real scalar and an imaginary scalar vanish separately, which the four terms can do in many combinations. The paper lists four of them, in decreasing strength; all four were verified on random elements. - Strongest: all four halves mutually orthogonal. From any quaternion $p_1$ the three quaternions $p_2 = p_1e_1$, $p_3 = p_1e_2$ and $p_4 = p_1e_3$ are mutually orthogonal, and the pair $\tilde{A} = p_1 + ip_2$, $\tilde{B} = p_3 + ip_4$ is orthogonal; both factors are then divisors of zero, equal-norm orthogonal halves being exactly the null condition. - Weaker I: the real halves are orthogonal to each other and the imaginary halves to each other, so the real scalar vanishes term by term, while the two cross terms are equal and opposite, so the imaginary scalar cancels. Every biquaternion $\tilde{A}$ gives such a partner, $\tilde{B} = \tilde{A}e_1$ (or $\tilde{A}e_2$, or $\tilde{A}e_3$). - Weaker II: the real halves are not orthogonal to each other, and neither are the imaginary halves, but the two scalar products agree, sign included, so the real part cancels; the real part of each factor is orthogonal to the imaginary part of the other, so the imaginary part vanishes term by term. Here $\tilde{B} = \tilde{A}ie_1$ with $\tilde{A}$ arbitrary. - Weakest: no pair of halves is orthogonal, and both scalars cancel between the terms. Here $\tilde{B} = \tilde{A}(e_1 + ie_2)$, again with $\tilde{A}$ arbitrary, and the factor $e_1+ie_2$ is a divisor of zero, so $\tilde{B}$ is one as well. This is the grade in which the pairing vanishes for a reason unconnected with any pair of directions being perpendicular.

The vanishing is not the null condition. The strongest grade produces divisor-of-zero pairs, but rescaling the two halves destroys the divisor-of-zero property and leaves the orthogonality untouched. Verified: $\tilde{A} = e_0 + ie_1$ and $\tilde{B} = e_2 + ie_3$ are orthogonal with $N(\tilde{A}) = N(\tilde{B}) = 0$, while $\tilde{A} = e_0 + 2.5\,ie_1$ and $\tilde{B} = e_2 + 0.4\,ie_3$ are still orthogonal with $N(\tilde{A}) = -5.25$. Orthogonality of the pairing and membership in the null cone are therefore independent conditions, linked only in the strongest grade above. This is worth stating because the corpus uses "orthogonal" in two other senses (idempotents multiplying to zero, and the Hermitian complement) with no metric content, and because a null element is orthogonal to itself, $B(\tilde{Q},\tilde{Q}) = \mathrm{Sc}\,N(\tilde{Q}) = 0$, a case the criterion covers as well.

An open point. The paper records that the geometric reading of the pairing is left incomplete: with both factors complex, the moduli and the angle are complex, so there is no direct analogue of $|A||B|\cos\theta$ with a real angle, and the interpretation "requires further work". In the corpus's terms the numbers $N(\tilde{A})$ and $N(\tilde{B})$ are complex, no real angle between two biquaternions is defined in general, and the vanishing of the pairing is an algebraic condition rather than a statement about perpendicular real directions.

The Hermitian pairing

Hermitian conjugation defines a second pairing, $$ \langle \tilde{A},\tilde{B}\rangle_2 := 2\,\mathrm{Sc}\bigl(\tilde{A}^{*}\tilde{B}\bigr) , $$ which is sesquilinear and positive definite, and the Gram matrix on any real basis is positive definite. It is the pairing in which operators have adjoints, and it is not the biquaternion norm: the biquaternion norm is bilinear and indefinite, the Hermitian pairing is sesquilinear and definite, and the companion article distinguishes them for exactly this reason. Both are preserved by the appropriate group. The bilinear form is invariant under reversal-conjugation, $$ B\bigl(\tilde{\Lambda}^{\natural}\tilde{A}\tilde{\Lambda}^{\natural},\ \tilde{\Lambda}^{\natural}\tilde{B}\tilde{\Lambda}^{\natural}\bigr) = B(\tilde{A},\tilde{B}) \qquad \text{for unit-norm } \tilde{\Lambda} , $$ while the Hermitian form is invariant under the unitary action $\tilde{A}\mapsto \tilde{U}\tilde{A}\tilde{U}^{-1}$ with $\tilde{U}^{*} = \tilde{U}^{-1}$; the two statements are the Lorentz and unitary faces of the same algebra, and both were checked on random elements.

Adjoints and the Lattice

The reason the lattice matters for the relativistic equations is that the adjoint of a multiplication operator is read from ${}^{*}$, and the adjoint of a differential operator from the interplay of ${}^{*}$ with the derivative. Both statements are one line each.

Multiplication. On the Hermitian module with pairing $\langle\cdot,\cdot\rangle_2$, left multiplication by $\tilde{A}$ has adjoint left multiplication by $\tilde{A}^{*}$, $$ \langle \tilde{A}\tilde{F},\tilde{G}\rangle_2 = \langle \tilde{F},\tilde{A}^{*}\tilde{G}\rangle_2 , $$ because $\mathrm{Sc}((\tilde{A}\tilde{F})^{*}\tilde{G}) = \mathrm{Sc}(\tilde{F}^{*}\tilde{A}^{*}\tilde{G})$; the same holds for right multiplication, $R_{\tilde{A}}^{*} = R_{\tilde{A}^{*}}$. Verified on random biquaternions and random module elements. This is the statement that the Hermitian conjugation of the algebra is the operator adjoint of its multiplication, and it is what makes ${}^{*}$ the right involution for physical observables.

Differentiation. For the gradient of the operator article, $$ \tilde{\nabla} = \sum_\mu e_\mu\partial_\mu , \qquad \tilde{\nabla}^{*} = -\tilde{\nabla}^{\natural} , $$ the sign coming from the anti-self-adjointness of the partial derivatives and the conjugation of the basis, $e_\mu^{*} = -e_\mu$ for $\mu \neq 0$. The algebra identity used is $$ \sum_\mu \bigl(\partial_\mu\tilde{F}\bigr)^\dagger e_\mu = \bigl(\tilde{\nabla}^{\natural}\tilde{F}\bigr)^\dagger , $$ which is antilinearity of ${}^{*}$ and $(e_\mu^{*})^{*} = e_\mu$, verified on random complex coefficients. Composing, $$ \Box^{*} = \bigl(\tilde{\nabla}\tilde{\nabla}^{\natural}\bigr)^\dagger = \bigl(-\tilde{\nabla}\bigr)\bigl(-\tilde{\nabla}^{\natural}\bigr) = \Box , $$ so the d'Alembertian is self-adjoint because the reversal enters the two factors with the same sign twice. This is the precise sense in which "the operator article's adjoints rely on the involution lattice": the reversal ${}^{\natural}$ is the conjugate appearing in $\tilde{\nabla}^{*} = -\tilde{\nabla}^{\natural}$, the Hermitian conjugation ${}^{*}$ provides the adjoint operation itself, and the central sign of $\flat$ is the same sign that makes the first-order kernel material-valued.

The real structure. The fixed space of $\flat$ is the sector in which the first-order kernels and the material four-vectors live; the companion Antilinear Structure and the Two Kinds of Mass in Biquaternionic Form records $\flat$ as the algebra's real structure and uses it for the two kinds of mass. Here the only claim is structural: $\flat$ is an involution whose fixed space is $\mathbb{M}_-$, it is the sign-twisted Hermitian conjugation, and it is the unique one of the four maps that is neither a plain automorphism nor a plain anti-automorphism of the algebra, carrying instead the central twist $(\tilde{A}\tilde{B})^\flat = -\tilde{B}^\flat\tilde{A}^\flat$.

Summary

The biquaternion algebra carries four conjugations. Quaternion conjugation (reversal) ${}^{\natural}$ is the $\mathbb{C}$-linear anti-automorphism $e_k \mapsto -e_k$, $i \mapsto i$; complex conjugation $\bar{\cdot}$ is the $\mathbb{C}$-antilinear automorphism $i \mapsto -i$, $e_k\mapsto e_k$; Hermitian conjugation ${}^{*} = {}^{\natural}\circ\bar{\cdot}$ is the $\mathbb{C}$-antilinear anti-automorphism negating both; and the anti-Hermitian conjugation $\flat = -{}^{*}$ is the involution that fixes the material sector and satisfies $(\tilde{A}\tilde{B})^\flat = -\tilde{B}^\flat\tilde{A}^\flat$.

The four maps form the Klein four-group $\{\mathrm{id},{}^{\natural},\bar{\cdot},{}^{*}\}$ with ${}^{*} = {}^{\natural}\circ\bar{\cdot}$, their composition table verified on random elements; together with the central scalars $\{\pm1,\pm i\}$ they generate a group of order sixteen, $\mathbb{Z}_4\rtimes V_4 \cong D_8\times\mathbb{Z}_2$ (non-abelian; the direct product $\mathbb{Z}_4\times V_4$ is excluded by the centre), whose sixteen elements were enumerated and found distinct. Their eigenspaces are the framework's six distinguished subspaces, named one by one: the center subspace $\mathbb{C}_{\mathbb{B}}$ (dim $2$), the vector subspace $\mathrm{Vect}(\mathbb{B})$ (dim $6$), the quaternion subspace $\mathbb{H}_{\mathbb{B}}$ (dim $4$), the antiquaternion subspace $i\mathbb{H}_{\mathbb{B}}$ (dim $4$), the informational sector $\mathbb{M}_+$ (dim $4$) and the material sector $\mathbb{M}_-$ (dim $4$). The four fixed spaces are $\mathbb{C}_{\mathbb{B}}$, $\mathbb{H}_{\mathbb{B}}$, $\mathbb{M}_+$ and $\mathbb{M}_-$, with the intersection lattice tabulated: $\mathbb{C}_{\mathbb{B}} = (\mathbb{C}_{\mathbb{B}}\cap\mathbb{M}_+)\oplus(\mathbb{C}_{\mathbb{B}}\cap\mathbb{M}_-)$ and $\mathbb{H}_{\mathbb{B}} = (\mathbb{H}_{\mathbb{B}}\cap\mathbb{M}_+)\oplus(\mathbb{H}_{\mathbb{B}}\cap\mathbb{M}_-)$, and $\mathbb{M}_+\cap\mathbb{M}_- = \{0\}$. Every conjugation preserves each sector; multiplication by $\pm i$ exchanges them, and the operations that exchange the sectors are the eight-element set $m_{\pm i}V_4$, not the central multiplications alone.

Reversal defines the bilinear pairing $B(\tilde{A},\tilde{B}) = \mathrm{Sc}(\tilde{A}^{\natural}\tilde{B})$, Lorentzian on $\mathbb{M}_-$ with signature $(-,+,+,+)$ and positive definite on $\mathbb{H}_{\mathbb{B}}$; Hermitian conjugation defines the positive-definite sesquilinear pairing $2\,\mathrm{Sc}(\tilde{A}^{*}\tilde{B})$. Multiplication operators have adjoints $L_{\tilde{A}}^{*} = L_{\tilde{A}^{*}}$, and the gradient has adjoint $-\tilde{\nabla}^{\natural}$, so that $\Box^{*} = \Box$. Reversal is a word in the products, $\tilde{Q}^{\natural} = -\tfrac12(\tilde{Q} + e_1\tilde{Q}e_1 + e_2\tilde{Q}e_2 + e_3\tilde{Q}e_3)$, whereas complex conjugation is primitive and no word in $\tilde{Q}$ computes it; an element commutes with its reversal, $\tilde{Q}\tilde{Q}^{\natural} = \tilde{Q}^{\natural}\tilde{Q} = N(\tilde{Q})$, but need not commute with its complex conjugate, for which $\tilde{Q}\tilde{Q}^* - \tilde{Q}^*\tilde{Q} = 2i\,(q_iq_r - q_rq_i)$ vanishes only for quaternions, for imaginary biquaternions, and for co-planar elements. The four characters of $V_4$ are four projections onto the four real components, the scalar, bivector, vector and pseudoscalar of the geometric-algebra dictionary, and the bilinear pairing vanishes in four grades, its vanishing being independent of the biquaternion norm.

The lattice is not decoration: it is the set of real forms on which the biquaternion norm, the observables, the adjoints, and the material kernels all live.

Summary of Notation

Further Reading

• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2001), for reversion, Clifford conjugation, the grade involution, and the classification of the anti-involutions of the Clifford algebras.
• Ian R. Porteous, Clifford Algebras and the Classical Groups (Cambridge, 1995), for the real forms of the complexified algebras and the correspondence between conjugations and real structures.
• F. Reese Harvey, Spinors and Calibrations (Academic Press, 1990), for the quaternionic and complex structures on $\mathbb{H}$ and $M_2(\mathbb{C})$.
• Israel M. Gel'fand and Mikhail A. Naimark, Unitäre Darstellungen der klassischen Gruppen (Akademie-Verlag, 1957), for the invariant Hermitian and bilinear forms on the quaternion algebra and the unitary and Lorentz groups preserving them.
• Claude Chevalley, The Algebraic Theory of Spinors (Columbia, 1954), for the anti-automorphisms of the Clifford algebra and the biquaternion norm built from reversal.
• Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1 (Cambridge, 1984), for the two-component spinor conjugation structures and the correspondence with the quaternion conjugation.
• S. J. Sangwine, T. A. Ell, and N. Le Bihan, "Fundamental representations and algebraic properties of biquaternions or complexified quaternions", Advances in Applied Clifford Algebras 21 (2011) 607–636, for the reversal word, the non-commutation of a biquaternion with its complex conjugate, the four graded projections, and the criterion for the vanishing of the bilinear pairing.
• Conventions in the Biquaternion Universe and Relations Between Subspaces, the companion articles, for the conventions the four conjugations are named by, for the six distinguished subspaces named one by one with their closure properties, and for the six-subspace intersection table of which the four-space lattice here is the restriction.