Biquaternion Automorphisms and Derivations

Introduction

The biquaternion algebra is the complexification of the quaternion algebra,

$$ \mathbb{B} = \mathbb{C} \otimes_{\mathbb{R}} \mathbb{H}. $$

It carries two structures, distinguished by the ground field. Over $\mathbb{C}$ it is a four-dimensional central simple algebra; over $\mathbb{R}$ the same set is an eight-dimensional algebra, simple but not central. This article describes two standard invariants of $\mathbb{B}$: the group of algebra automorphisms and the Lie algebra of derivations. Both depend on the ground field, so the two views are kept separate and the field is named at each step.

We use the conventions of Biquaternion Algebra throughout: the basis $\{e_0, e_1, e_2, e_3\}$ with $e_k^2 = -e_0$; the central scalar imaginary $i$ with $i^2 = -1$; the conjugations ${}^{\natural}$, $\bar{\cdot}$, ${}^{*} = {}^{\natural} \circ \bar{\cdot}$, ${}^{\flat} = -{}^{*}$; and the six distinguished subspaces $\mathbb{C}_{\mathbb{B}}$, $\mathrm{Vect}(\mathbb{B})$, $\mathbb{H}_{\mathbb{B}}$, $i\mathbb{H}_{\mathbb{B}}$, $\mathbb{M}_+$, $\mathbb{M}_-$. A general element is $\tilde{Q} = Q_0 e_0 + Q_1 e_1 + Q_2 e_2 + Q_3 e_3$ with $Q_\mu \in \mathbb{C}$.

No new results are claimed; everything below is standard structure theory of the algebra over each of the two ground fields.

Physically the two groups are the two kinds of symmetry of the framework. The inner automorphisms are the proper orthochronous motions — they are exactly the rotations and boosts of Biquaternion Rotations and Lorentz Transformations, so the connected component of $\operatorname{Aut}_{\mathbb{C}}(\mathbb{B})$ is the Lorentz group of the material sector. The conjugate-linear outer coset is the discrete symmetry that no motion can supply: parity and time reversal, the operations that exchange the two chiralities and the two sectors. The derivations are the infinitesimal generators of the same motions, the Lie algebra whose commutators are the angular-momentum algebra of Biquaternion Lie Algebra.

Standing Facts: Simplicity and the Centre

The automorphism group and the derivation algebra of $\mathbb{B}$ are governed by two structural facts, recorded here and used throughout. Both are proved elsewhere and are cited, not reproved.

Simplicity. Over $\mathbb{C}$, the only two-sided ideals of $\mathbb{B}$ are $0$ and $\mathbb{B}$: the algebra is simple. The proof is in Biquaternion Ideals and Peirce Decomposition, §2. The Two-Sided Ideals: Simplicity of $\mathbb{B}$, where the ideal structure of the algebra is developed; the lattice is the same over $\mathbb{R}$ and over $\mathbb{C}$, because the condition $\tilde R\tilde T,\tilde T\tilde R\in I$ is field-independent.

The centre. The centre of $\mathbb{B}$ is $Z(\mathbb{B})=\mathbb{C}_{\mathbb{B}}=\mathbb{C}e_0$, a complex vector space of dimension $1$, the subspace $\mathbb{C}_{\mathbb{B}}$ of Biquaternion Algebra. Hence $\mathbb{B}$ is central simple over $\mathbb{C}$ — simple, of dimension $4$, with centre exactly $\mathbb{C}$ — while over $\mathbb{R}$ the same algebra is simple but not central, its centre $\mathbb{C}_{\mathbb{B}}\cong\mathbb{C}$ being strictly larger than $\mathbb{R}e_0$. Stating that "$\mathbb{B}$ is central simple" without naming the field is false over $\mathbb{R}$, and the sections below respect the distinction.

Automorphisms over $\mathbb{C}$

Throughout this section the ground field is $\mathbb{C}$.

Definition. A $\mathbb{C}$-algebra automorphism of $\mathbb{B}$ is a bijective $\mathbb{C}$-linear map $\sigma : \mathbb{B} \to \mathbb{B}$ with $\sigma(\tilde R\tilde T) = \sigma(\tilde R)\sigma(\tilde T)$ and $\sigma(e_0) = e_0$. These maps form a group under composition, written $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$.

Inner automorphisms. For any invertible $g \in \mathbb{B}$ the map $\iota_g(\tilde R) = g \tilde R g^{-1}$ is a $\mathbb{C}$-algebra automorphism, the inner automorphism determined by $g$. Since $\mathbb{C}$ is central, $\iota_g = \iota_{\lambda g}$ for every nonzero $\lambda \in \mathbb{C}$, so $\iota_g$ depends only on the class of $g$ modulo the scalars.

Theorem (Skolem–Noether). For any field $k$ and $n \geq 1$, every $k$-algebra automorphism of $M_n(k)$ is inner: for each $\sigma$ there exists $g \in GL_n(k)$ with $\sigma(\tilde R) = g\tilde Rg^{-1}$ for all $\tilde R$.

Applied to $\mathbb{B}$, which is central simple over $\mathbb{C}$, the theorem says that every $\mathbb{C}$-linear automorphism of $\mathbb{B}$ is inner: each has the form $\iota_g$ for an invertible $g \in \mathbb{B}^{\times}$. Since $\iota_g$ depends only on $g$ modulo the central scalars, there is a surjection

$$ \mathbb{B}^{\times} \longrightarrow \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}), \qquad g \longmapsto \iota_g, $$

whose kernel is the group of nonzero central scalars $\mathbb{C}^{\times} e_0$. Hence

$$ \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \cong \mathbb{B}^{\times} / \mathbb{C}^{\times} .$$

Dimension. Since $\mathbb{B}^{\times}$ has complex dimension $4$ and the central scalars have complex dimension $1$, $\dim_{\mathbb{C}} \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) = 4 - 1 = 3$. Equivalently, the group has real dimension $6$, and it is connected.

Corollary (the projective linear group). Under the matrix model $\mathbb{B}\cong M_2(\mathbb{C})$ the units are $GL(2,\mathbb{C})$ and the nonzero central scalars are the scalar matrices, so $$ \operatorname{Aut}_{\mathbb{C}}(\mathbb{B})\cong\mathbb{B}^{\times}/\mathbb{C}^{\times}\cong PGL(2,\mathbb{C}), $$ the projective general linear group, of complex dimension three. This is the biquaternion form of the classical isomorphism $\operatorname{Aut}(M_n(k))\cong PGL(n,k)$.

Corollary (the norm). Every $\mathbb{C}$-linear automorphism preserves the biquaternion norm, and every $\mathbb{R}$-linear automorphism preserves the real norm $r=\sqrt{|N|}$. For an inner automorphism the complex statement is multiplicativity, $N(g\tilde{Q}g^{-1})=N(g)N(\tilde{Q})N(g)^{-1}=N(\tilde{Q})$; complex conjugation reverses the sign of the imaginary part, $N(\tilde{Q}^{*})=N(\tilde{Q})^{*}$, and so preserves $|N|$ without preserving $N$ itself.

Remark (automorphisms against isometries). The automorphism group is a proper subgroup of the full isometry group $O(N)$ of the real norm on $\mathbb{B}\cong\mathbb{R}^8$: $PGL(2,\mathbb{C})$ has real dimension $6$, whereas the isometry group of a non-degenerate form of signature $(4,4)$ on $\mathbb{R}^8$ has real dimension $\tfrac{8\cdot7}{2}=28$. The automorphisms are the isometries that also preserve the algebra; the further isometries are not algebra maps.

Example. For $g = e_1$, with $e_1^{-1} = -e_1$, conjugation fixes $e_1$, $e_0$, $i$ and reverses the signs of $e_2$ and $e_3$: $\iota_{e_1}(e_1) = e_1$, $\iota_{e_1}(e_2) = -e_2$, $\iota_{e_1}(e_3) = -e_3$. Indeed $e_1 e_2 e_1^{-1} = -(e_1 e_2)e_1 = -e_3 e_1 = -e_2$, using $e_1 e_2 = e_3$ and $e_3 e_1 = e_2$.

Remark. Quaternion conjugation satisfies $(\tilde R\tilde T)^{\natural} = \tilde T^{\natural}\,\tilde R^{\natural}$ and is an anti-automorphism, not an automorphism, so it is not in $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$; complex conjugation is an automorphism but is not $\mathbb{C}$-linear, so it is not in $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$ either. It reappears over $\mathbb{R}$ below.

Automorphisms over $\mathbb{R}$

Now the ground field is $\mathbb{R}$: $\mathbb{B}$ is eight-dimensional, and automorphisms need only be $\mathbb{R}$-linear, not $\mathbb{C}$-linear, which makes the group strictly larger.

Automorphisms preserve the center. If $\sigma$ is an $\mathbb{R}$-algebra automorphism and $z$ is central, then $\sigma(z)\sigma(\tilde R) = \sigma(z\tilde R) = \sigma(\tilde Rz) = \sigma(\tilde R)\sigma(z)$ for every $\tilde R$, so $\sigma(z)$ is central. Thus $\sigma$ restricts to an $\mathbb{R}$-algebra automorphism of $\mathbb{C}_{\mathbb{B}} \cong \mathbb{C}$; since $\mathbb{C}$ as a real algebra has exactly two automorphisms, the identity and $\kappa(z) = z^{*}$, restriction gives a homomorphism $\rho : \mathrm{Aut}_{\mathbb{R}}(\mathbb{B}) \to \mathrm{Aut}_{\mathbb{R}}(\mathbb{C}_{\mathbb{B}}) = \{\mathrm{id}, \kappa\} \cong \mathbb{Z}/2$.

The kernel is the $\mathbb{C}$-linear part. An automorphism lies in $\ker \rho$ exactly when it fixes the center pointwise, and an $\mathbb{R}$-linear map fixing $\mathbb{C}_{\mathbb{B}}$ pointwise is automatically $\mathbb{C}$-linear, since it commutes with multiplication by the central element $i$. Hence $\ker \rho = \mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$, the group computed above; these are precisely the inner automorphisms, by Skolem–Noether.

The conjugation coset. The map $\rho$ is surjective: $c(\tilde{Q}) = \bar{\tilde{Q}}$ is an $\mathbb{R}$-algebra automorphism, since $(\tilde R\tilde T)^{*} = \tilde R^{*}\tilde T^{*}$, and it induces $\kappa$ on the center, since $c(i) = -i$. It is not $\mathbb{C}$-linear, and it is not inner, because inner automorphisms fix the center pointwise while $c(i) = -i \neq i$. So the extension is nontrivial.

The full real automorphism group. There is a short exact sequence $1 \to \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \to \mathrm{Aut}_{\mathbb{R}}(\mathbb{B}) \xrightarrow{\rho} \mathbb{Z}/2 \to 1$, split by $c$ because $c^{2} = \mathrm{id}$. Hence

$$ \mathrm{Aut}_{\mathbb{R}}(\mathbb{B}) \cong \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \rtimes \mathbb{Z}/2, $$

the generator acting on $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$ by $\iota_g \mapsto \iota_{g^{*}}$. Concretely, every real automorphism of $\mathbb{B}$ has exactly one of the two forms $\sigma(\tilde R) = g \tilde R g^{-1}$ or $\sigma(\tilde R) = g\, \tilde R^{*}\, g^{-1}$, with $g \in \mathbb{B}^{\times}$ determined up to a nonzero complex scalar. The first family is the identity coset of $\mathbb{C}$-linear inner automorphisms; the second is the coset of $c$, consisting of conjugate-linear automorphisms.

Dimension and scope. As a real Lie group, $\mathrm{Aut}_{\mathbb{R}}(\mathbb{B})$ has real dimension $6$ and exactly two connected components, each a copy of $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$; it is larger and is a real group rather than a complex one. Skolem–Noether describes the identity component $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$; the conjugate-linear coset exists because $\mathbb{C}/\mathbb{R}$ has a nontrivial Galois automorphism, and it is not inner.

Derivations over $\mathbb{C}$

Throughout this section the ground field is $\mathbb{C}$.

Definition. A $\mathbb{C}$-linear derivation of $\mathbb{B}$ is a $\mathbb{C}$-linear map $D : \mathbb{B} \to \mathbb{B}$ with $D(\tilde R\tilde T) = D(\tilde R)\,\tilde T + \tilde R\,D(\tilde T)$ for all $\tilde R, \tilde T \in \mathbb{B}$. The set of all such maps is a complex vector space, written $\mathrm{Der}_{\mathbb{C}}(\mathbb{B})$, and it is a Lie algebra under the commutator bracket $[D_1, D_2] = D_1 \circ D_2 - D_2 \circ D_1$.

Every derivation satisfies $D(e_0) = 0$, since $D(e_0) = D(e_0 e_0) = 2D(e_0)$.

Inner derivations. For each $\tilde A \in \mathbb{B}$ the map $\mathrm{ad}_{\tilde A}(\tilde R) = \tilde A \tilde R - \tilde R \tilde A = [\tilde A, \tilde R]$ is a $\mathbb{C}$-linear derivation, the inner derivation determined by $\tilde A$; the Jacobi identity in the form $[[\tilde A,\tilde R],\tilde T] + [\tilde R,[\tilde A,\tilde T]] = [\tilde A,[\tilde R,\tilde T]]$ is exactly the Leibniz rule for $\mathrm{ad}_{\tilde A}$. The map $\mathrm{ad} : \mathbb{B} \to \mathrm{Der}_{\mathbb{C}}(\mathbb{B})$, $\tilde A \mapsto \mathrm{ad}_{\tilde A}$, is $\mathbb{C}$-linear with kernel the center, since $\mathrm{ad}_{\tilde A} = 0$ says exactly that $\tilde A$ commutes with everything: $\ker(\mathrm{ad}) = Z(\mathbb{B}) = \mathbb{C}_{\mathbb{B}}$.

Every derivation is inner. For a central simple algebra over a field every derivation is inner. Hence $\mathrm{ad}$ is surjective and induces a $\mathbb{C}$-linear isomorphism

$$ \mathrm{Der}_{\mathbb{C}}(\mathbb{B}) \cong \mathbb{B} / \mathbb{C}_{\mathbb{B}}. $$

Identification with the traceless part. The quotient $\mathbb{B}/\mathbb{C}_{\mathbb{B}}$ is the traceless part of $\mathbb{B}$, the subspace of vanishing scalar part $\{\tilde{Q} : Q_0 = 0\} = \mathrm{span}_{\mathbb{C}}\{e_1, e_2, e_3\}$, the complex pure-vector part of $\mathbb{B}$, a complex vector space of dimension $3$. The isomorphism is one of Lie algebras, because $[\mathrm{ad}_{\tilde A}, \mathrm{ad}_{\tilde B}] = \mathrm{ad}_{[\tilde A,\tilde B]}$.

Dimension. The derivation space has $\dim_{\mathbb{C}} \mathrm{Der}_{\mathbb{C}}(\mathbb{B}) = 3$, hence real dimension $6$. This matches the automorphism group, which also has complex dimension $3$; its Lie algebra is $\mathrm{Der}_{\mathbb{C}}(\mathbb{B})$.

The bivector reading. Under the identification $\mathbb{B} \cong \mathrm{Cl}^{+}_{1,3}$, the real span $\mathrm{span}_{\mathbb{R}}\{e_1, e_2, e_3, i e_1, i e_2, i e_3\}$ is the six-dimensional bivector subspace, the real form of the complex pure-vector part, and its elements bracket into themselves. Thus the derivation algebra, viewed as a real Lie algebra, is the bivector part of $\mathrm{Cl}^{+}_{1,3}$, equivalently, as a real Lie algebra, the realification of the complex pure-vector part, of real dimension $6$, which is the Lie algebra of the Lorentz group. The traceless part, the complex pure-vector part, and the bivector part are three descriptions of one object, with $\dim_{\mathbb{R}} = 2 \dim_{\mathbb{C}}$.

An explicit basis. The three derivations $D_1 = \tfrac{1}{2}\mathrm{ad}_{e_1}$, $D_2 = \tfrac{1}{2}\mathrm{ad}_{e_2}$, $D_3 = \tfrac{1}{2}\mathrm{ad}_{e_3}$ are $\mathbb{C}$-linear, so each vanishes on $e_0$ and on $i$, and they act on $e_1, e_2, e_3$ by

$$ D_1(e_2) = e_3, \quad D_1(e_3) = -e_2; \qquad D_2(e_3) = e_1, \quad D_2(e_1) = -e_3; \qquad D_3(e_1) = e_2, \quad D_3(e_2) = -e_1, $$

with $D_1(e_1) = D_2(e_2) = D_3(e_3) = 0$. They satisfy the standard commutation relations,

$$ [D_1, D_2] = D_3, \qquad [D_2, D_3] = D_1, \qquad [D_3, D_1] = D_2, $$

so $\{D_1, D_2, D_3\}$ is a complex basis of $\mathrm{Der}_{\mathbb{C}}(\mathbb{B})$. For instance $D_1(e_2) = \tfrac{1}{2}(e_1 e_2 - e_2 e_1) = \tfrac{1}{2}(e_3 + e_3) = e_3$, and $[D_1, D_2] = \tfrac{1}{4}\mathrm{ad}_{[e_1,e_2]} = \tfrac{1}{4}\mathrm{ad}_{2e_3} = D_3$.

Derivations and automorphisms. The two structures are linked by the exponential: $\exp(t\,\mathrm{ad}_{\tilde A})(\tilde R) = e^{t\tilde A}\, \tilde R\, e^{-t\tilde A}$ for $\tilde A \in \mathbb{B}$, $t \in \mathbb{R}$. The right-hand side is the inner automorphism determined by $e^{t\tilde A}$, so the Lie algebra of the automorphism group is the derivation algebra, $\mathrm{Lie}\,\mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) = \mathrm{Der}_{\mathbb{C}}(\mathbb{B})$.

Derivations over $\mathbb{R}$

Now the ground field is $\mathbb{R}$. An $\mathbb{R}$-linear derivation is required to satisfy the Leibniz rule but need not be $\mathbb{C}$-linear. At first sight this seems to allow a larger space, but in fact it does not.

Every real derivation is automatically $\mathbb{C}$-linear. Let $D$ be an $\mathbb{R}$-linear derivation. As in the automorphism case, $D$ maps the center into itself: if $z$ is central, then for every $\tilde R$,

$$ D(z)\tilde R = D(z\tilde R) - zD(\tilde R) = D(\tilde Rz) - D(\tilde R)z = \tilde RD(z). $$

So $D$ restricts to a derivation $\mathbb{C}_{\mathbb{B}} \to \mathbb{C}_{\mathbb{B}}$. But $\mathbb{C}$ has no nonzero $\mathbb{R}$-linear derivations: a derivation of $\mathbb{C}$ is determined by $D(i)$, and $0 = D(-1) = D(i^{2}) = i\,D(i) + D(i)\,i = 2i\,D(i)$ forces $D(i) = 0$, hence $D$ vanishes on the center. Since $i$ is central, the Leibniz rule then gives $D(i \tilde R) = D(i)\,\tilde R + i\,D(\tilde R) = i\,D(\tilde R)$, so $D$ is $\mathbb{C}$-linear. Therefore $\mathrm{Der}_{\mathbb{R}}(\mathbb{B}) = \mathrm{Der}_{\mathbb{C}}(\mathbb{B})$.

There is no semilinear analogue for derivations: a derivation cannot conjugate a coefficient.

Dimension and structure. Consequently the real derivation space has real dimension $6$, that is, complex dimension $3$, and $\mathrm{Der}_{\mathbb{R}}(\mathbb{B}) = \mathrm{Der}_{\mathbb{C}}(\mathbb{B})$, the realification of the complex derivation algebra. As a real Lie algebra this is the orthogonal Lie algebra $\mathrm{SO}(1,3)$ of the Lorentz group, equivalently the bivector subspace of $\mathrm{Cl}^{+}_{1,3}$ under the commutator. The derivations $D_1, D_2, D_3$ of the previous section span it over $\mathbb{C}$, and together with $iD_1, iD_2, iD_3$ over $\mathbb{R}$.

Summary of the asymmetry. For automorphisms, the real group is strictly larger than the complex one, because complex conjugation supplies a second coset. For derivations, the real and complex spaces coincide, because the center is étale over $\mathbb{R}$ and admits no nonzero derivation. This is a ground-field distinction and not a convention.

Worked Examples

The conjugation involution. Complex conjugation $c(\tilde{Q}) = \bar{\tilde{Q}}$ is an $\mathbb{R}$-algebra automorphism with $c(e_k) = e_k$ ($k = 0,1,2,3$) and $c(i) = -i$; it fixes the quaternion subspace $\mathbb{H}_{\mathbb{B}}$ pointwise, negates the scalar imaginary, satisfies $c^{2} = \mathrm{id}$, preserves the product, and is conjugate-linear over $\mathbb{C}$. It is not inner, because inner automorphisms fix the center pointwise whereas $c(i) = -i$, so it represents the nontrivial coset of $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$ in $\mathrm{Aut}_{\mathbb{R}}(\mathbb{B})$.

A rotation derivation. Take $D = D_3 = \tfrac{1}{2}\mathrm{ad}_{e_3}$, so that $D(e_1) = e_2$ and $D(e_2) = -e_1$. Its exponential acts by

$$ \exp(tD)(e_1) = \cos t\, e_1 + \sin t\, e_2, $$

as one checks by summing the series. Equivalently, $e^{t e_3/2} = \cos(t/2) + \sin(t/2) e_3$, and conjugation by this unit quaternion rotates the $e_1$-$e_2$ plane, matching $\exp(t D) = \mathrm{Ad}_{e^{t e_3/2}}$.

Every element of $\mathrm{span}_{\mathbb{C}}\{e_1, e_2, e_3\}$ gives an inner derivation, and $\mathrm{ad}_{\tilde A}$ depends only on $\tilde A$ modulo the center $\mathbb{C}_{\mathbb{B}}$; for instance $\tilde A = e_1 + i e_2$ gives a derivation that is not a scalar multiple of any $D_k$. Physical reading: symmetries finite and infinitesimal. The inner automorphisms are the proper orthochronous motions: they are exactly the rotations and boosts of Biquaternion Rotations and Lorentz Transformations, so the connected component of $\operatorname{Aut}_{\mathbb{C}}(\mathbb{B})$ is the Lorentz group of the material sector. The conjugate-linear outer coset is the discrete symmetry that no motion can supply — parity and time reversal, the operations that exchange the two chiralities and the two sectors $\mathbb{M}_\pm$. The derivations are the infinitesimal generators of the same motions: $\mathrm{Der}(\mathbb{B})\cong\mathbb{B}/\mathbb{C}_{\mathbb{B}}$ is the Lorentz Lie algebra, whose commutators are the angular-momentum algebra of Biquaternion Lie Algebra, and the exponential of a derivation returns to an automorphism exactly when the motion is a rotation or a boost.

Summary

The two ground fields give the following table; the field is stated explicitly in every entry.

Structure	Over $\mathbb{C}$	Over $\mathbb{R}$
Algebra	simple $\mathbb{C}$-algebra, complex dimension $4$	simple $\mathbb{R}$-algebra, real dimension $8$
Ideals	$\{0\}$ and $\mathbb{B}$ only	$\{0\}$ and $\mathbb{B}$ only
Center	$\mathbb{C}_{\mathbb{B}} = \mathbb{C} e_0$, dimension $1$ over $\mathbb{C}$	$\mathbb{C}_{\mathbb{B}} \cong \mathbb{C}$, dimension $2$ over $\mathbb{R}$
Central simple?	yes, central simple over $\mathbb{C}$	no: simple, but center $\mathbb{C} \neq \mathbb{R}$
Automorphism group	$\mathbb{B}^{\times}/\mathbb{C}^{\times}$, complex dimension $3$ ($6$ over $\mathbb{R}$), connected	$\mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \rtimes \mathbb{Z}/2$, real dimension $6$, two components
Derivation space (the Lie algebra of the automorphism group)	$\mathrm{Der}_{\mathbb{C}}(\mathbb{B}) \cong \mathbb{B}/\mathbb{C}_{\mathbb{B}}$, complex dimension $3$ ($6$ over $\mathbb{R}$); traceless part, complex pure-vector part, bivectors	$\mathrm{Der}_{\mathbb{R}}(\mathbb{B}) = \mathrm{Der}_{\mathbb{C}}(\mathbb{B})$, real dimension $6$, isomorphic to the Lorentz algebra $\mathrm{SO}(1,3)$

In summary: over $\mathbb{C}$ the algebra is central simple, every automorphism is inner, and $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \cong \mathbb{B}^{\times}/\mathbb{C}^{\times}$; over $\mathbb{R}$ complex conjugation adds a second, conjugate-linear coset, so $\mathrm{Aut}_{\mathbb{R}}(\mathbb{B}) \cong \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \rtimes \mathbb{Z}/2$ is strictly larger; and over either field the derivations are the inner derivations $\tilde R \mapsto [\tilde A,\tilde R]$ with $\tilde A$ traceless, forming $\mathbb{B}/\mathbb{C}_{\mathbb{B}}$, of dimension $3$ over $\mathbb{C}$ and $6$ over $\mathbb{R}$, identified with the bivector part of $\mathrm{Cl}^{+}_{1,3}$. Physically the inner automorphisms are the proper orthochronous motions, the conjugate-linear outer coset carries parity and time reversal, and the derivations are the infinitesimal generators of the Lorentz algebra.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C} \otimes_{\mathbb{R}} \mathbb{H}$	Biquaternion algebra; complex dimension $4$, real dimension $8$
$e_0, e_1, e_2, e_3$	Algebra basis, $e_0 = 1$, $e_k^2 = -e_0$
$i$	Central scalar imaginary, $i^2 = -1$
$\mathbb{C}_{\mathbb{B}} = \mathrm{span}_{\mathbb{R}}\{e_0, ie_0\}$	Center of $\mathbb{B}$; the scalar subspace
$\mathrm{Vect}(\mathbb{B})$	Vector subspace, vanishing scalar part
$\mathbb{H}_{\mathbb{B}}, i\mathbb{H}_{\mathbb{B}}$	Real-quaternion and anti-quaternion subspaces
$\mathbb{M}_+, \mathbb{M}_-$	Hermitian and anti-Hermitian subspaces
${}^{\natural}, \bar{\cdot}, {}^{*}, {}^{\flat}$	Quaternion, complex, Hermitian and anti-Hermitian conjugations
$\mathrm{Aut}_{\mathbb{C}}(\mathbb{B})$	Algebra automorphisms over $\mathbb{C}$, $\cong \mathbb{B}^{\times}/\mathbb{C}^{\times}$
$\mathrm{Aut}_{\mathbb{R}}(\mathbb{B})$	Algebra automorphisms over $\mathbb{R}$, $\cong \mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \rtimes \mathbb{Z}/2$
$\mathrm{Der}_{\mathbb{C}}(\mathbb{B})$	Derivations over $\mathbb{C}$, $\cong \mathbb{B}/\mathbb{C}_{\mathbb{B}}$
$\mathrm{Der}_{\mathbb{R}}(\mathbb{B})$	Derivations over $\mathbb{R}$, $\cong \mathrm{SO}(1,3)$ as a real Lie algebra
$\mathrm{ad}_{\tilde A}(\tilde R) = [\tilde A,\tilde R]$	Inner derivation by $\tilde A$
$D_k = \tfrac{1}{2}\mathrm{ad}_{e_k}$	Basis of $\mathrm{Der}_{\mathbb{C}}(\mathbb{B})$, $[D_1,D_2] = D_3$ etc.
$\mathrm{Cl}_{1,3}^{+}$	Even Clifford algebra; isomorphic to $\mathbb{B}$