The Lie–Jordan Decomposition of a Bilinear Product and the Jordan Triple System

Introduction

A single bilinear product on a vector space carries two structures at once. Its symmetrisation $x \bullet y = \tfrac12(xy + yx)$ is commutative, and its antisymmetrisation $[x,y] = \tfrac12(xy - yx)$ is anticommutative. The corpus treats the two separately: the symmetric product is the subject of Jordan Algebras and of the Symmetric Linear Algebras of this part, and the antisymmetric bracket is the subject of Lie Algebras and of the Anti-symmetric Linear Algebras. This article records the observation — Liu Yu-Fen's, in the source of the biquaternion dirac construction — that for the specific product built from the Dirac matrices the two structures are both present and compatible, and that the ternary product they generate is a Jordan triple system.

The setting is the biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H} \cong M_2(\mathbb{C})$, read as a complex four-dimensional vector space $V = \mathbb{C}^4$ with a bilinear composition whose structure constants are the source's

$$ c^{\mu\nu\lambda} = \bigl(t^{\mu\nu\lambda\rho} - i\varepsilon^{\mu\nu\lambda\rho}\bigr)k_\rho, $$

where $t^{\mu\nu\lambda\rho} = \tfrac14\mathrm{tr}(\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$ and $\varepsilon^{\mu\nu\lambda\rho} = \tfrac{i}{4}\mathrm{tr}(\gamma_5\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$ are the gamma traces and $k^\mu$ is a neutral vector. The composition

$$ (G \star H)^\lambda = G_\mu\, c^{\mu\lambda\nu} H_\nu $$

is associative and has $k$ as its unit; it is the biquaternion product in a particular basis. The point of this article is what that product is, algebraically: one bilinear map, three structures.

The Two Halves of One Product

Definitions

For $G, H \in V$ put

$$ G \bullet H = \tfrac12\bigl(G \star H + H \star G\bigr), \qquad [G, H] = \tfrac12\bigl(G \star H - H \star G\bigr). $$

Then $G \bullet H = H \bullet G$ and $[G,H] = -[H,G]$: the symmetrisation is commutative, the antisymmetrisation anticommutative. Both are bilinear, and either, together with the other, reconstructs the product: $G \star H = G \bullet H + [G, H]$. The two are therefore not two structures on $V$ but the two halves of one structure.

The Lie Half

The antisymmetrisation satisfies the Jacobi identity

$$ [G,[H,K]] + [H,[K,G]] + [K,[G,H]] = 0, $$

so $(V,[\cdot,\cdot])$ is a Lie algebra. Because the product that defines it is associative, this is the commutator algebra of an associative algebra: the Lie algebra associated to $M_2(\mathbb{C})$, namely $\mathfrak{gl}(2,\mathbb{C}) \cong \mathfrak{sl}(2,\mathbb{C}) \oplus \mathbb{C}$. The corpus's Lie Algebras and the biquaternion Lie algebra article record the general construction; here it is the first half of the source's decomposition, and it is the half that the neutral element $j$ (the spacelike unit) leans on, since the antisymmetric symbol $\varepsilon$ is what survives the antisymmetrisation.

The Jordan Half

The symmetrisation satisfies the Jordan identity

$$ G \bullet (H \bullet (G \bullet G)) = (G \bullet H) \bullet (G \bullet G), $$

so $(V, \bullet)$ is a Jordan algebra. Because the product is associative, this is the special Jordan algebra $M_2(\mathbb{C})^+$ — the symmetrisation of an associative algebra, which is the general source of special Jordan algebras, and which here is the corpus's $H_2(\mathbb{C})$ with the Hermitian part singled out. The corpus's Jordan Algebras and Special and Exceptional Jordan Algebras record the theory; the second half of the source's decomposition lands exactly there.

The Compatibility

The two halves are not independent: the bracket is a derivation of the Jordan product,

$$ [G, H \bullet K] = [G,H] \bullet K + H \bullet [G,K], $$

and the sum

$$ [G, H \bullet K] + [H, K \bullet G] + [K, G \bullet H] = 0 $$

vanishes identically. The first identity says the Lie algebra acts on the Jordan algebra by derivations; the second is the fundamental identity that the source names, the compatibility condition that makes the pair a Lie–Jordan algebra in the sense of the literature on the Kantor–Koecher construction. The three identities — Jacobi, Jordan, and this compatibility — are what the source's "revised and corrected" account of its algebra asserts; the correction it makes to its earlier formulation is exactly a sign, the sign of the $\varepsilon$ term in $c^{\mu\nu\lambda}$, without which the antisymmetric half fails Jacobi and the symmetric half fails Jordan.

The Jordan Triple System

The Ternary Product

The symmetrised product generates a ternary operation,

$$ \{G, H, K\} = (G \bullet H) \bullet K + (K \bullet H) \bullet G - (G \bullet K) \bullet H, $$

which is the linearisation of the quadratic representation $U_G H = G \bullet (H \bullet G)$ of the Jordan algebra. Equivalently, and this is the form the source's construction makes natural, the ternary product of an associative algebra with an involution is

$$ \{G, H, K\} = (G \star H) \star K + (K \star H) \star G . $$

Both forms are ternary operations on $V$, and both are symmetric in the outer two arguments, $\{G,H,K\} = \{K,H,G\}$.

The Triple Identities

A Jordan triple system is a vector space with a ternary product that is linear in each argument, symmetric in the outer two, and satisfies the five-linear Jordan triple identity

$$ \{x, y, \{u, v, w\}\} = \{\{x,y,u\}, v, w\} - \{u, \{y,x,v\}, w\} + \{u, v, \{x,y,w\}\}. $$

Both ternary products above satisfy this identity. A Jordan triple system need not come from a Jordan algebra — the triple identity is weaker than the Jordan identity — so the ternary structure is the more general object, and the source's construction lands on it: the cubic form of the triality construction ($A_\mu\bar\Psi\gamma^\mu\Psi$, in the companion physics articles) is the ternary product read with the vector space as the third factor. This is why the source presents its algebra as an order-$\ell$ ding, of which the order-three part is the Jordan triple system.

The Block Structure

The four-dimensional carrier has one more piece of visible structure. In the trinomial basis the units $e_\mu = i\hat e_\mu$ are $2\times2$ block matrices: each unit is a $2\times2$ matrix of $2\times2$ blocks, and the blocks of the three spatial units close under multiplication into the quaternion group $Q_8$ modulo its centre. The source groups these blocks under the name periodic matrices $J_i$; the block form itself is what the algebra supplies, and it exhibits $\mathbb{B} \cong M_2(\mathbb{C})$ directly — the algebra is a $2\times2$ matrix algebra, its symmetric part is the special Jordan algebra, and its skew part is the Lie algebra. The verified content of the block structure is exactly the quaternion relations $\hat e_k^2 = -I$ and $\hat e_1\hat e_2 = \hat e_3$; the identification of the individual blocks with the source's $J_i$ is the source's, and the corpus records it as such.

What Is Verified and What Is the Source's

Verified on the corpus's convention, each identity on random arguments, $\le 100$ per identity, throwaway script outside the repository:

Jacobi for the antisymmetric half;
the Jordan identity for the symmetric half;
the derivation identity $[G, H\bullet K] = [G,H]\bullet K + H\bullet[G,K]$;
the fundamental identity $[G,H\bullet K] + [H,K\bullet G] + [K,G\bullet H] = 0$;
the symmetry $\{G,H,K\} = \{K,H,G\}$ and the five-linear Jordan triple identity, for both ternary forms;
that the composition is associative with unit $k$, and that the blocks of $e_1, e_2, e_3$ reproduce the quaternion relations.

The source's, transcribed and not independently derivable from the identities above: the naming of the block matrices $J_i$ and the claim that the "revised" account corrects an earlier formulation by a sign; the identification of the ternary product with the source's order-three ding; and the general definition of an order-$\ell$ ding. The identities themselves are the definitional ones for Lie, Jordan and Jordan-triple structures; what the source adds is the observation that the same product carries all three, and the explicit construction that shows it.

A caution. That Jacobi and Jordan hold is, for an associative algebra, automatic: the commutator of an associative algebra is always a Lie bracket and its symmetrisation always a Jordan algebra. The content of the article is therefore not the holding of the identities but the identification of the algebra: the structure constants $c^{\mu\nu\lambda}$ with the normed condition are those of $M_2(\mathbb{C})$, so the Lie half is $\mathfrak{gl}(2,\mathbb{C})$, the Jordan half is $M_2(\mathbb{C})^+$, and the triple system is the Jordan triple system of the same matrix algebra. The source's correction matters because a wrong sign in the $\varepsilon$ term produces a different product — one that is neither associative nor normed — and then none of the three structures is the standard one.

Open Questions

The Kantor–Koecher construction. A Lie–Jordan algebra with a compatibility condition is the input to the Kantor–Koecher construction, which builds a Lie algebra (and hence a graded Lie algebra) from a Jordan algebra. Is the source's algebra a Kantor pair, and is the resulting Lie algebra $\mathfrak{sl}(2,\mathbb{C})$ itself, or a larger one?
The order-three structure. The algebra carries an order-three automorphism (the triality of the companion physics articles). Does that automorphism act on the Lie half, the Jordan half and the triple system in the way the $S_3$ outer automorphism acts on the three representations of $\mathrm{Spin}(8)$? The triple structure — one Lie algebra, one Jordan algebra, one Jordan triple — is suggestive, and the question is whether the suggestion is a theorem.
The exceptional case. The biquaternion algebra is 4-dimensional, and its Jordan half is special. The first exceptional Jordan algebra, the Albert algebra, needs 27 dimensions and the octonions. Does the order-$\ell$ ding construction have an octonionic lifting, and does the lifting land on the exceptional structures of Special and Exceptional Jordan Algebras?
The grading. The compatibility identity is the germ of a $\mathbb{Z}/2$-graded Lie algebra structure, with the bracket as the odd part acting on the even part. The corpus's Superalgebras and Graded Structures treats the $\mathbb{Z}/2$ grading directly; whether the source's Lie–Jordan pair is best read as a graded Lie algebra is open.

These questions are open.

Summary

A bilinear product on a vector space has two halves: its symmetrisation $G\bullet H = \tfrac12(G\star H + H\star G)$ and its antisymmetrisation $[G,H] = \tfrac12(G\star H - H\star G)$. For the product built from the Dirac matrices, $c^{\mu\nu\lambda} = (t^{\mu\nu\lambda\rho} - i\varepsilon^{\mu\nu\lambda\rho})k_\rho$, the antisymmetric half satisfies the Jacobi identity and is the Lie algebra $\mathfrak{gl}(2,\mathbb{C})$, and the symmetric half satisfies the Jordan identity and is the special Jordan algebra $M_2(\mathbb{C})^+$. The two halves are compatible: the bracket is a derivation of the Jordan product, $[G,H\bullet K] = [G,H]\bullet K + H\bullet[G,K]$, and the fundamental identity $[G,H\bullet K] + [H,K\bullet G] + [K,G\bullet H] = 0$ holds. The symmetrised product generates a ternary operation $\{G,H,K\} = (G\bullet H)\bullet K + (K\bullet H)\bullet G - (G\bullet K)\bullet H$, symmetric in its outer arguments and satisfying the five-linear Jordan triple identity; the source's order-three ding is this triple system. The block structure of the units $e_\mu$ exhibits the matrix algebra directly, with the four blocks of each unit being the source's periodic matrices $J_i$.

The identities Jacobi, Jordan, the derivation identity, the fundamental identity and the Jordan triple identity are all verified. The content of the construction is the identification: one product, the structure constants of $M_2(\mathbb{C})$, carries three compatible algebraic structures at once.

Summary of Notation

Symbol	Meaning
$V = \mathbb{C}^4$	Carrier of the algebra, $\mathbb{B} \cong M_2(\mathbb{C})$
$G, H, K$	Elements of $V$
$G \star H$	The composition, $(G\star H)^\lambda = G_\mu c^{\mu\lambda\nu}H_\nu$
$c^{\mu\nu\lambda} = (t^{\mu\nu\lambda\rho} - i\varepsilon^{\mu\nu\lambda\rho})k_\rho$	Structure constants
$t^{\mu\nu\lambda\rho} = \tfrac14\mathrm{tr}(\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$	Symmetric gamma trace
$\varepsilon^{\mu\nu\lambda\rho} = \tfrac{i}{4}\mathrm{tr}(\gamma_5\gamma^\mu\gamma^\nu\gamma^\lambda\gamma^\rho)$	Totally antisymmetric symbol
$k^\mu$	Neutral vector, the unit of the composition
$G \bullet H = \tfrac12(G\star H + H\star G)$	Jordan (symmetric) product
$[G,H] = \tfrac12(G\star H - H\star G)$	Lie (antisymmetric) bracket
$\{G,H,K\}$	Ternary (Jordan triple) product
$e_\mu = i\hat e_\mu$	Units of the trinomial basis, $\hat e_k^2 = -I$
$J_i$	The source's name for the four $2\times2$ blocks of the units