Special and Exceptional Jordan Algebras

Introduction

This article analyses the Jordan algebras that come from associative algebras and the one that does not. The base structure is a commutative ring $R$ with identity $1 \neq 0$, and the Jordan conventions — the commutative product $\bullet$, the square $x^2 = x \bullet x$, the multiplication operators $L_x$, the Jordan identity $[L_x, L_{x^2}] = 0$ — are those fixed in Jordan Algebras. The general setting is that article; here we study the dichotomy and the examples.

A Jordan algebra is special if it embeds into the symmetrisation $A^+$ of an associative algebra $A$ and exceptional if it does not. The classification of the finite-dimensional formally real algebras, quoted in Jordan Algebras, lists three families: the matrix algebras $H_n(D)$ over the associative composition algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ with $n \geq 3$, the degree-two spin factors, together with the one-dimensional algebra $\mathbb{R}$, all of which are special; and the single algebra $H_3(\mathbb{O})$ of Hermitian $3 \times 3$ matrices over the octonions, the Albert algebra, which is exceptional. That one exception is the whole reason the subject is a theory of its own rather than a chapter of associative algebra.

The article constructs the special algebras $H_n(D)$ and verifies that they are Jordan algebras, constructs the octonions by Cayley–Dickson doubling, assembles the Albert algebra and records its basic invariants, and states the exceptionality theorem with the finite identity — Glennie's — that witnesses it. The spin factors and their Clifford envelopes are in The Clifford, Pin and Spin Groups with Signed Inner Conjugation.

Special and Exceptional

The Definition

Let $A$ be an associative $R$-algebra and let $A^+$ be the same module with the symmetrised product $x \bullet y = \tfrac{1}{2}(xy + yx)$, defined when $2$ is invertible, or $xy + yx$ in general; by Jordan Algebras, $A^+$ is a Jordan algebra. A Jordan algebra $J$ is special if there exist an associative algebra $A$ and an injective Jordan homomorphism $J \hookrightarrow A^+$; it is exceptional otherwise. A special identity, or $s$-identity, is a polynomial identity in the operations $\bullet$ that holds in every $A^+$.

The notion of specialness is not defined by a property of the multiplication table alone, because "embeds in some $A^+$" quantifies over an algebra that does not appear in the data. This is the source of the difficulty: to prove a Jordan algebra exceptional one must show that no associative algebra whatsoever can receive it.

The Free Special Jordan Algebra

Let $X$ be a set. The free associative algebra $R\langle X\rangle$ is the tensor algebra of the free module on $X$ (Tensor Powers and the Free Algebra); its elements are non-commutative polynomials. The free special Jordan algebra $\operatorname{SJ}(X)$ is the Jordan subalgebra of $R\langle X\rangle^+$ generated by $X$, that is, the set of symmetrised non-commutative polynomials in the variables $X$.

The free Jordan algebra $\operatorname{J}(X)$ is defined by the universal property that Jordan homomorphisms $\operatorname{J}(X) \to J$ correspond to arbitrary maps $X \to J$. There is a canonical surjection

$$ \operatorname{J}(X) \twoheadrightarrow \operatorname{SJ}(X) , $$

because $X$ generates a Jordan subalgebra of $R\langle X\rangle^+$. The question of whether this map is an isomorphism is the question of whether every Jordan algebra is special, and it was settled in the negative.

Theorem (Cohn, Shirshov). The canonical map $\operatorname{J}(X) \to \operatorname{SJ}(X)$ is an isomorphism for $|X| \leq 2$ and is not an isomorphism for $|X| \geq 3$. Consequently every Jordan algebra generated by two elements is special, and there are Jordan algebras generated by three elements that are not.

Glennie's Identity

The failure for three generators is witnessed by an explicit $s$-identity.

Theorem (Glennie). There is a homogeneous polynomial identity of degree $8$ in the Jordan operations, the Glennie identity, that holds in every special Jordan algebra and fails in the free Jordan algebra on three generators.

The identity is

$$ G(x,y,z) = 0, $$

where $G$ is a homogeneous polynomial of degree $8$, an alternating sum of monomials in the operators $L_x, L_y, L_z$; the precise expression is recorded by Glennie (1963) and reproduced in the standard references. Its existence proves at once that $\operatorname{J}(X) \neq \operatorname{SJ}(X)$ for $|X| \geq 3$, and hence that some Jordan algebra is exceptional. The Albert algebra below is the concrete witness.

Hermitian Matrices

Conjugation and Hermitian Matrices

Let $D$ be a composition algebra over $R$: a unital $R$-algebra with a conjugation $u \mapsto u^{\natural}$ such that

$$ u u^{\natural} = u^{\natural} u \in R, \qquad (uv)(uv)^{\natural} = (u u^{\natural})(v v^{\natural}), $$

with $u + u^{\natural} \in R$ for all $u$. The classical examples are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$; the first three are associative and $\mathbb{O}$ is not. For a matrix $x$ over $D$, let $x^*$ denote the conjugate transpose. The Hermitian matrices are

$$ H_n(D) = \{x \in M_n(D) : x^* = x\}, $$

the module of matrices fixed by the involution $x \mapsto x^*$. When $D$ is associative, $H_n(D)$ is closed under the symmetrised product $\tfrac{1}{2}(xy + yx)$: if $x^* = x$ and $y^* = y$, then

$$ \tfrac{1}{2}(xy + yx)^* = \tfrac{1}{2}(y^* x^* + x^* y^*) = \tfrac{1}{2}(yx + xy), $$

so the product is again Hermitian. Thus $H_n(D)$ is a Jordan subalgebra of $M_n(D)^+$.

The Jordan Identity for $H_n(D)$

Theorem. For $D$ associative and $2$ invertible, $H_n(D)$ is a Jordan algebra. In particular $H_n(\mathbb{R})$, $H_n(\mathbb{C})$ and $H_n(\mathbb{H})$ are Jordan algebras for every $n \geq 1$.

Proof. $H_n(D)$ is a submodule of $M_n(D)$ closed under $\bullet$, and a subalgebra of a Jordan algebra is a Jordan algebra, since the identity is verified in the larger algebra. The ambient $M_n(D)^+$ is a Jordan algebra by the theorem of Jordan Algebras on $A^+$.

Remark. Closure under $\bullet$ uses only that conjugation is an anti-automorphism, so it does not need associativity. For $D = \mathbb{O}$ conjugation is still an anti-automorphism and closure still holds; what fails is the associativity of $M_n(\mathbb{O})$ used in the proof of the Jordan identity for $A^+$. The case $n = 3$ survives, and that is the Albert algebra.

Matrix Units and the Peirce Decomposition

Assume $2$ is invertible and let $e_{ii} \in H_n(D)$ be the diagonal matrix with $1$ in the $i$-th place. The $e_{ii}$ are pairwise orthogonal idempotents with $\sum_i e_{ii} = 1$, and the Peirce decomposition (Jordan Algebras) with respect to $e = e_{11}$ is

$$ H_n(D) = J_1(e) \oplus J_{1/2}(e) \oplus J_0(e), \qquad J_1(e) = \{u \in D : u^{\natural} = u\}\,e_{11}, \quad J_0(e) = H_{n-1}(D), \quad J_{1/2}(e) = \{\text{the off-diagonal entries of the first row and column}\}. $$

This decomposition is the tool with which the structure of $H_n(D)$ is reduced to that of $H_{n-1}(D)$ by induction. For $n = 2$ the middle space is the module $D$ of off-diagonal entries, and the algebra has degree two in the sense that every element satisfies a quadratic minimal polynomial; it is a spin factor. For $n \geq 3$ the algebra has degree $n$ and is not a spin factor.

Dimensions

The dimension of $H_n(D)$ over $R$ is

$$ \dim_R H_n(D) = n \cdot \dim_R \{u \in D : u^{\natural} = u\} + \binom{n}{2} \dim_R D , $$

the $n$ diagonal entries lying in the fixed field of the conjugation and the $\binom{n}{2}$ off-diagonal entries each contributing a copy of $D$. For a composition algebra the fixed field is $R\cdot1$, of dimension $1$, so $\dim_R H_n(D) = n + \binom{n}{2}\dim_R D$; for $D = \mathbb{R}, \mathbb{C}, \mathbb{H}$ this gives

$$ \dim H_n(\mathbb{R}) = \frac{n(n+1)}{2}, \qquad \dim H_n(\mathbb{C}) = n^2, \qquad \dim H_n(\mathbb{H}) = n(2n - 1). $$

Example. $H_2(\mathbb{R})$ has dimension $3$ and is the smallest spin factor; $H_2(\mathbb{C})$ has dimension $4$; $H_2(\mathbb{H})$ has dimension $6$; $H_3(\mathbb{R})$ has dimension $6$; $H_3(\mathbb{H})$ has dimension $15$. These values are consistent with the formula and are the classical Jordan algebras of Hermitian matrices.

The Octonions

Cayley–Dickson Doubling

The octonions are obtained from the quaternions by one doubling. Write $\mathbb{H}$ for the quaternions with basis $1, e_1, e_2, e_3$ and conjugation $\bar{}$, and put

$$ \mathbb{O} = \mathbb{H} \oplus \mathbb{H}\ell, \qquad \ell^2 = -1, \qquad \ell x = x^{\natural}\,\ell \quad (x \in \mathbb{H}), $$

so that the pair $(a, b)$ is written $a + b\ell$; the multiplication, for $a, b, c, d \in \mathbb{H}$, is the Cayley–Dickson product

$$ (a + b\ell)(c + d\ell) = (ac - d^{\natural}\, b) + (da + b c^{\natural})\,\ell . $$

The relation $\ell x = x^{\natural}\ell$, which follows from the product, says that $\ell$ anticommutes with the imaginary quaternions $e_1, e_2, e_3$; because $\mathbb{O}$ is not associative, the product is not an associative expansion of the notation $a + b\ell$.

The conjugation is $\overline{a + b\ell} = a^{\natural} - b\ell$, so $uu^{\natural} = a a^{\natural} + b b^{\natural} \in \mathbb{R}$. The eight elements $1, e_1, e_2, e_3, \ell, e_1\ell, e_2\ell, e_3\ell$ form a basis, so $\dim_{\mathbb{R}} \mathbb{O} = 8$.

Theorem. $\mathbb{O}$ is a composition algebra: $(uv)(uv)^{\natural} = (u u^{\natural})(vv^{\natural})$ and $(uv) v^{\natural} = u(vv^{\natural})$ for all $u, v$. It is alternative, satisfying $u(uv) = (uu)v$ and $(vu)u = v(uu)$, and it is not associative.

The verification is a direct computation in the eight basis elements; the model above is the one used throughout. Non-associativity is the property that makes the Albert algebra exceptional, and alternativity is the property that keeps it a Jordan algebra.

Coordinates

An octonion is written $u = u_0 + \sum_{i=1}^{7} u_i \iota_i$ with $u_i \in \mathbb{R}$ and $\iota_i$ the seven imaginary basis elements $\iota_1 = e_1$, $\iota_2 = e_2$, $\iota_3 = e_3$, $\iota_4 = \ell$, $\iota_5 = e_1\ell$, $\iota_6 = e_2\ell$, $\iota_7 = e_3\ell$. Conjugation negates the imaginary part, $u^{\natural} = u_0 - \sum_i u_i\iota_i$, and $u u^{\natural} = \sum_{i=0}^{7} u_i^2$. The real part is $\operatorname{Re}(u) = \tfrac{1}{2}(u + u^{\natural}) = u_0$, and the imaginary part is $\operatorname{Im}(u) = \tfrac{1}{2}(u - u^{\natural})$.

The Albert Algebra

Definition

The Albert algebra is the $27$-dimensional real Jordan algebra

$$ H_3(\mathbb{O}) = \{x \in M_3(\mathbb{O}) : x^* = x\} $$

of Hermitian $3 \times 3$ matrices over $\mathbb{O}$, with the symmetrised product $x \bullet y = \tfrac{1}{2}(xy + yx)$.

A Hermitian $3 \times 3$ matrix has real diagonal entries (an octonion fixed by conjugation is real) and three independent off-diagonal octonion entries, the lower triangle being the conjugate transpose of the upper. Hence

$$ \dim_{\mathbb{R}} H_3(\mathbb{O}) = 3 + 3 \cdot 8 = 27 . $$

This is the value quoted in Jordan Algebras, and the algebra is the unique exceptional simple formally real Jordan algebra.

The Jordan Identity

Theorem (Albert). $H_3(\mathbb{O})$ with the symmetrised product is a Jordan algebra.

Proof (sketch). Let $x = \sum_{i,j} x_{ij} \otimes E_{ij}$ be a generic Hermitian matrix, with $x_{ii} \in \mathbb{R}$ and $x_{ji} = x^{\natural}_{ij}$. The product $x \bullet y = \tfrac{1}{2}(xy + yx)$ is again Hermitian, by the anti-automorphism property of the octonion conjugation, so $\bullet$ is a commutative product on $H_3(\mathbb{O})$. The Jordan identity $[L_x, L_{x^2}] = 0$ becomes, after expanding both sides over the eight basis octonions, a polynomial identity of degree four in the entries $x_{ij}$. The reduction uses the alternative laws $u(uv) = (uu)v$ and $(vu)u = v(uu)$, the vanishing of the real part of the associator, $\operatorname{Re}[u,v,w] = 0$, and the composition law $(uv)(uv)^{\natural} = (uu^{\natural})(v v^{\natural})$ with $u u^{\natural}$ real, which together bring the two sides to the same normal form. This is the classical theorem of Albert; the finite ingredient is the multiplication table of the eight basis units.

Remark. The same computation fails for $n \geq 4$: the symmetrised product on $H_n(\mathbb{O})$ is not a Jordan product for $n \geq 4$. The reason is that the proof of the Jordan identity for $A^+$ uses the associativity of the entries, which an associative composition algebra ($\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$) supplies and which the octonions supply only up to the alternative laws. The case $n = 3$ escapes because the Hermitian condition pairs every entry with its conjugate transpose, and the associator of octonions has vanishing real part, so the alternative laws suffice for the products that occur; from $n = 4$ onwards the identity contains a product of three off-diagonal octonions whose bracketing matters.

Idempotents

The diagonal matrices

$$ e_1 = \operatorname{diag}(1, 0, 0), \quad e_2 = \operatorname{diag}(0, 1, 0), \quad e_3 = \operatorname{diag}(0, 0, 1) $$

are pairwise orthogonal idempotents with $e_1 + e_2 + e_3 = 1$, a Jordan frame of the algebra. The Peirce decomposition with respect to $e_1$ is

$$ H_3(\mathbb{O}) = J_1(e_1) \oplus J_{1/2}(e_1) \oplus J_0(e_1), $$

with $J_1(e_1) = \mathbb{R}e_1$ of dimension $1$, $J_0(e_1) = H_2(\mathbb{O})$ of dimension $10$, and $J_{1/2}(e_1)$ of dimension $16$, spanned by the two off-diagonal octonion entries of the first row and column. The dimensions add: $1 + 10 + 16 = 27$. The subalgebra $H_2(\mathbb{O})$ is a spin factor, and this is the route by which the exceptional algebra is built from the degree-two algebras.

Exceptionality

The Theorem

Theorem (Albert). The Albert algebra $H_3(\mathbb{O})$ is exceptional: there is no associative algebra $A$ and no injective Jordan homomorphism $H_3(\mathbb{O}) \hookrightarrow A^+$.

Proof (idea). Every special Jordan algebra satisfies Glennie's identity of degree $8$. One evaluates the Glennie polynomial on the general element of $H_3(\mathbb{O})$, expressed in the basis of the $27$ matrix units together with the eight octonion coordinates, and finds a nonzero result; a concrete choice of three elements suffices, and the evaluation is finite, involving only the multiplication table of $\mathbb{O}$. Hence $H_3(\mathbb{O})$ satisfies no such identity and cannot be special. The statement is classical; the modern form of the argument identifies the symmetric bilinear trace form and the cubic invariant as the obstructions, and shows that a special algebra of degree three with such an invariant admitting composition cannot exist.

Remark. The exceptionality is not a rare accident. In the classification over $\mathbb{R}$ the Albert algebra is the unique exceptional simple algebra, and it is exactly the case in which the composition algebra of coefficients is the non-associative one, with a matrix size of three; the algebras $H_n(\mathbb{O})$ are defined for every $n$, but the symmetrised product is a Jordan product only for $n \leq 3$.

The Automorphism Group

The automorphism group of the Albert algebra is the exceptional Lie group

$$ \operatorname{Aut}(H_3(\mathbb{O})) \cong F_4 , $$

of dimension $52$; its Lie algebra is the derivation algebra $\operatorname{Der}(H_3(\mathbb{O})) \cong \mathrm{F}_4$, also of dimension $52$. The derivations are exactly the Jordan derivations of the previous article, $\operatorname{Der}(H_3(\mathbb{O})) = \{\delta: \delta(x\bullet y) = \delta x \bullet y + x \bullet \delta y\}$; the trace form is invariant under them by the theorem of Jordan Algebras. The group $F_4$ acts transitively on the idempotents of rank one; the larger structure group is generated by $E_6$ of dimension $78$ together with the dilations $x \mapsto \lambda x$, whose multipliers $\lambda^3$ fill $\mathbb{R}^{\times}$, and it has dimension $79$.

The Tits–Kantor–Koecher Construction

From Jordan Algebras to Lie Algebras

The derivation algebra $\operatorname{Der}(J)$ of a Jordan algebra is a Lie algebra, but it is not the only one attached to $J$. The Tits–Kantor–Koecher (TKK) construction produces a Lie algebra from $J$ itself, and it does so functorially.

Theorem (Tits, Kantor, Koecher). To each Jordan algebra $J$ over a ring in which $2$ is invertible there is assigned functorially a Lie algebra $\mathrm{G}(J)$ admitting a three-term grading

$$ \mathrm{G}(J) = J^- \oplus \mathrm{G}_0 \oplus J^+ , $$

in which $J^+$ and $J^-$ are two copies of the module $J$ and are abelian, the middle piece $\mathrm{G}_0$ is generated by the multiplication operators $L_x$ and the inner derivations $[L_x, L_y]$, and the bracket is determined by the action of $\mathrm{G}_0$ on $J^\pm$ together with the rule for $[x^+, y^-]$ derived from the Jordan product. The algebra $\mathrm{G}(J)$ carries an involution exchanging $J^+$ and $J^-$, and the construction is natural in $J$.

The explicit bracket and the verification of the Jacobi identity are in the standard references. For the special algebras $J = H_n(D)$ the construction recovers the classical simple Lie algebras of the corresponding matrix groups; for the Albert algebra it gives the exceptional Lie algebra

$$ \mathrm{G}(H_3(\mathbb{O})) \cong \mathrm{E}_7 , $$

of dimension $133$, while

$$ \operatorname{Der}(H_3(\mathbb{O})) \cong \mathrm{F}_4, \qquad \mathrm{STR}(H_3(\mathbb{O})) \cong \mathrm{E}_6 \oplus \mathbb{R} , $$

where $\mathrm{STR}(J) = \operatorname{Der}(J) \oplus L(J)$ is the structure algebra, the Lie algebra of the structure group $\operatorname{Str}(J)$ of dimension $79$; its centre is spanned by $L_1$, which generates the dilations, its derived algebra is the invariant-preserving $\mathrm{E}_6$ of dimension $78$, and the graded pieces of $\mathrm{G}(J)$ add to $27 + 79 + 27 = 133$, as the dimension of $\mathrm{E}_7$ requires. The TKK construction is the cleanest bridge between the Jordan theory of this category and the Lie theory, and it shows that the exceptional Jordan algebra is not an isolated curiosity but the entry point to the exceptional series.

The Simple Algebras and Their Invariants

The simple formally real algebras of the classification are tabulated below with their degrees (the size of a Jordan frame), their dimensions, and the derivation algebra, which controls the automorphism group.

$J$	degree	$\dim_{\mathbb{R}} J$	$\operatorname{Der}(J)$
$\mathbb{R} = JSpin_0$	$1$	$1$	$0$
$JSpin_n$, $n \geq 2$	$2$	$n + 1$	$\mathrm{SO}(n)$
$H_n(\mathbb{R})$, $n \geq 3$	$n$	$\tfrac{n(n+1)}{2}$	$\mathrm{SO}(n)$
$H_n(\mathbb{C})$, $n \geq 3$	$n$	$n^2$	$\mathrm{SU}(n)$
$H_n(\mathbb{H})$, $n \geq 3$	$n$	$n(2n-1)$	$\mathrm{Sp}(n)$
$H_3(\mathbb{O})$	$3$	$27$	$\mathrm{F}_4$

The degree of a simple Jordan algebra is the number of orthogonal idempotents in a maximal Jordan frame; it is the invariant that separates the spin factors from the matrix algebras, and it is the size of the matrix algebra in the matrix family. The derivation algebras are the classical skew algebras for the matrix family and the exceptional algebra $\mathrm{F}_4$ for the Albert algebra; the table is the precise form of the classification statement of Jordan Algebras.

Summary

A Jordan algebra is special if it embeds in the symmetrisation $A^+$ of an associative algebra and exceptional otherwise. The canonical surjection $\operatorname{J}(X)\twoheadrightarrow\operatorname{SJ}(X)$ from the free Jordan algebra to the free special Jordan algebra is an isomorphism for at most two generators and is not one for three or more (Cohn, Shirshov); the failure is witnessed by Glennie's identity, a special identity of degree $8$. The Hermitian matrix algebras $H_n(D)$ over the associative composition algebras $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ are special by the $A^+$ theorem, with dimensions $\tfrac{n(n+1)}{2}$, $n^2$ and $n(2n-1)$ respectively, and Peirce decompositions that reduce them inductively to smaller $n$. The octonions, constructed by Cayley–Dickson doubling from $\mathbb{H}$, form a non-associative alternative composition algebra of dimension $8$. The Albert algebra $H_3(\mathbb{O})$ of Hermitian $3\times3$ octonionic matrices is a Jordan algebra of dimension $27$; its symmetrised product satisfies the Jordan identity by Albert's theorem, the alternative laws of $\mathbb{O}$ together with the vanishing of the real part of the associator bringing the two sides of $[L_x, L_{x^2}] = 0$ to the same normal form, and the same computation fails from size $4$ onwards. It is exceptional (Albert), it is the unique exceptional simple formally real Jordan algebra, and its automorphism group is the exceptional Lie group $F_4$ of dimension $52$.

Summary of Notation

Symbol	Meaning
$R$	Commutative ring with identity $1 \neq 0$
$A^+$	Symmetrisation of the associative algebra $A$
Special / exceptional	Embeds / does not embed in some $A^+$
$s$-identity	Polynomial identity valid in every $A^+$
$\operatorname{J}(X)$, $\operatorname{SJ}(X)$	Free Jordan algebra, free special Jordan algebra on $X$
$D$	Composition algebra: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$
$u \mapsto u^{\natural}$, $x \mapsto x^*$	Conjugation, conjugate transpose
$H_n(D)$	Hermitian $n \times n$ matrices over $D$
$\mathbb{O} = \mathbb{H}\oplus\mathbb{H}\ell$	Octonions by Cayley–Dickson doubling, $\ell^2 = -1$
$\iota_1, \ldots, \iota_7$	Imaginary octonion basis units
$T(x)$, $S(x)$, $N(x)$	Trace and the quadratic and cubic invariants on $H_3(\mathbb{O})$
$x^{\#} = x^2 - T(x)x + S(x)1$	Quadratic adjoint of $x$
$e_1, e_2, e_3$	Jordan frame of $H_3(\mathbb{O})$
$\delta$	A derivation, $\delta(x\bullet y) = \delta x \bullet y + x \bullet \delta y$
$F_4$, $\mathrm{F}_4$	Automorphism group and derivation algebra of $H_3(\mathbb{O})$
$\mathrm{G}(J)$	Tits–Kantor–Koecher Lie algebra of $J$
$\mathrm{STR}(H_3(\mathbb{O})) \cong \mathrm{E}_6 \oplus \mathbb{R}$	Structure algebra, $\operatorname{Der}(H_3(\mathbb{O})) \oplus L(H_3(\mathbb{O}))$
$\operatorname{Str}(H_3(\mathbb{O}))$	Structure group of the Albert algebra, of dimension $79$; the dilations $x \mapsto \lambda x$ extend the invariant-preserving $E_6$ of dimension $78$
$\mathrm{G}(H_3(\mathbb{O})) \cong \mathrm{E}_7$	Tits–Kantor–Koecher algebra of $H_3(\mathbb{O})$