Constructions of Exceptional Lie Algebras via Jordan Algebras

Lecture Notes: Prof. Efim Zelmanov on Jordan Algebras and Exceptional Lie Algebras (December 17, 2025).

Preliminary

The Cayley-Dickson Construction

The foundation begins with the sequence of division algebras:

$$\mathbb{R}\to \mathbb{C}\to \mathbb{H}\to \mathbb{O}$$

Remark that octonions are the largest normed division algebras, which is non-associative.

Derivations and Inner Derivations

Recall that we defined derivations before.

(ab)\mathcal{D} =(a\mathcal{D} )b+a(b\mathcal{D} ).

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Inner derivations are derivations constructed specifically from the multiplication operators of the algebra itself. 

Case 1: Lie Algebras  For a Lie algebra $L$, the inner derivation is defined by the adjoint action: ${\mathrm{ad}}_x(y):=[x,y]$. A classical result states that for any semi-simple Lie algebra, all derivations are inner, i.e., $\mathrm{Der}(L)=\mathrm{Inder}(L)$. 

Case 2: Non-associative / Jordan Algebras  For a non-associative algebra $A$ (such as a Jordan algebra), we typically define inner derivations using the commutators of multiplication operators. Let $L_x$ denote the left multiplication operator ($L_x(y)=xy$). We define the generator:

$$D_{a,b}:=[L_a,L_b]=L_a{L}_b-L_b{L}_a$$

By ^70d2f4, we know the span of all such operators

$$\mathrm{Inder}(A)=\mathrm{span}\{D_{a,b}\mid a,b\in A\}$$

is a Lie algebra.

Examples: from Jordan Algebra to Lie Algebra

Derivations of Octonions: $G_2$

Note that $\mathrm{Der}(\mathbb{O})=\mathrm{Inder}(\mathbb{O})=G_2$. The group of symmetries (automorphisms) of the Octonions forms the exceptional Lie group $G_2$, whose dimension is $14$.

Derivations of the Albert Algebra: $F_4$

Recall the Albert algebra

where $\alpha ,\beta ,\gamma \in \mathbb{R}$ and $a,b,c\in \mathbb{O}$. Hence, we know $\dim {\mathbb{H}}_3(\mathbb{O})=27$.

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The trace of $M$ is $t(M)=\alpha +\beta +\gamma$. Although the algebra is non-associative, the trace form is associative, i.e. $t((xy)z)=t(x(yz))$.

This invariance is crucial for the algebraic structure. The derivations (infinitesimal symmetries) of the Albert Algebra form the exceptional Lie algebra $F_4$, that is,

$$\mathrm{Der}\left(H_3(\mathbb{O})\right)=\mathrm{Inder}(H_3(\mathbb{O}))=F_4,$$

where inner derivations is defined by commutators of multiplication operators

$$D_{a,b}=\frac{1}{4}\left[R_a,R_b\right]$$

and its dimension is $52$.

From Lie Algebra to Jordan Algebra: the TKK Construction (Tits-Kantor-Koecher)

Assume the Lie algebra $L$ contains an ${\mathfrak{sl}}_2$-triple $\{e,f,h\}$ satisfying

$$[e,f]=h,[h,e]=2e,[h,f]=-2f.$$

The adjoint action of $h$, i.e. $\mathrm{ad}(h)$ induces a short grading (3-grading) on $L$

$$L=L_{-2}\oplus L_0\oplus L_2,$$

where $L_k=\{x\in L:[h,x]=kx\}$.

Define the Jordan product as follows. We focus on the space $L_2$ (the “top” component), and take an element $e\in L_{-2}$ (from the “bottom” component) as a pivot. For any vectors $x,y\in L_2$, define a new product $\circ$

$$x\circ y:=[[x,e],y],$$

then the bracket $[x,e]$ lands in $L_0$, and $x\circ y\in L_2$.

J. Tits, 1962

• The pair $(L_2,\circ )$ forms a Jordan Algebra.
• This establishes a one-to-one correspondence between Jordan algebras and certain graded Lie algebras.

Remark. This explains why the “top corner” of the Exceptional Lie Algebras (like the $E_7$ structure acting on the 56-dimensional space) often reveals Jordan algebra structures.

Method 1: the TKK Algebra

Recall that

Define $U_{a,b}=\frac{1}{2}(U_{a+b}-U_a-U_b)$ and $\{axb\}=x{U}_{a,b}$ (Jordan triple product).

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In fact, the triple product of Jordan algebra is defined as

$$\{a,b,c\}:=(a\cdot b)\cdot c+a\cdot (b\cdot c)-(a\cdot c)\cdot b,$$

Furthermore, if the Jordan algebra is special, then $a\cdot b=\frac{1}{2}(ab+ba)$ and $\{a,b,c\}=\frac{1}{2}(abc+cba)$.

Motivation: Rectangular Matrices

Consider a linear space $M_{p,q}$ spanned by $p\times q$ matrices. We cannot multiply two such matrices directly, but we can define a triple product using the transpose $aba:=a\left(b^{T}a\right)$, and the corresponding triple product is

$$\{a,b,c\}=\frac{1}{2}(a{b}^{T}c+c{b}^{T}a).$$

It makes $M_{p,q}$ a Jordan triple system, even though it is not a Jordan Algebra.

Jordan Pairs $(P^{-},P^{+})$

Instead of a single space, we have a pair of spaces $(P^{-},P^{+})$ acting on each other via the triple product:

• $\left\{P^{+},P^{-},P^{+}\right\}\subseteq P^{+}$
• $\left\{P^{-},P^{+},P^{-}\right\}\subseteq P^{-}$

Constructing the Lie Algebra $L$ (the TKK Algebra)

Now we construct a Lie algebra from a Jordan pair.

Define $L$ as

$$L=\underset{\text{Bottom }}{\underbrace{P^{-}}}\oplus \underset{\text{Middle }\left(L_0\right)}{\underbrace{\delta \left(P^{-},P^{+}\right)}}\oplus \underset{\text{Top }}{\underbrace{P^{+}}}.$$

For $x^{-}\in P^{-}$ and $y^{+}\in P^{+}$, we define their Lie bracket to be an operator in the middle layer $[x^{-},y^{+}]:=\delta (x^{-},y^{+})$. Physically, $\delta (x^{-},y^{+})$ represents the inner derivation generated by these elements.

The middle layer acts on the top layer via Jordan triple product, that is, we define the Lie bracket as $\left[\delta \left(x^{-},y^{+}\right),z^{+}\right]:=-\left\{y^{+},x^{-},z^{+}\right\}$. A similar action applies to $P^{-}$.

Method 2: Tits Construction​

Generic Norms and Traces

Recall that Octonions are non-associative. We define the associator to measure this failure

$$(x,y,z):=(xy)z-x(yz).$$

Although associators of $\mathbb{O}$ and $H_3(\mathbb{O})$ are non-zero, we can still define generic trace and generic norm such that Cayley-Hamilton theorem holds.

Rank 2: The Octonions $\mathbb{O}$ is non-associative but alternative, and its trace satisfies $t(ab)=t(ba)$, it has a characteristic equation

$$x^2-t(x)x+n(x)1=0.$$

Rank 3: The Albert algebra $H_3(\mathbb{O})$ is a commutative (Jordan) algebra, and it has a characteristic equation

$$x^3-T(x)x^2+S(x)x-N(x)1=0.$$

The Trace-Zero Subspaces ($A^0$ and $J^0$)

To construct the Lie algebra, we need to decompose the algebras into their scalar parts (trace) and vector parts (trace-free).

For the Composition Algebra $A$ (Rank 2, e.g., $\mathbb{O}$), define the trace-zero subspace

$$A^0=\{a\in A\mid t(a)=0\},$$

and define a product $a\times b$ by projecting the standard multiplication onto the trace-zero space

$$a\times b=ab-\frac{1}{2}t(ab)1.$$

When $A=\mathbb{O}$, $A^0$ has dimension $7$.

For the Jordan Algebra $J$ (Rank 3, e.g., $H_3(\mathbb{O})$), Define the trace-zero subspace

$$J^0=\{x\in J\mid T(x)=0\}.$$

When $J={\mathbb{H}}_3(\mathbb{O})$, $J^0$ has dimension $26$.

Tits Construction of $E_8$

Define $A=\mathbb{O}$, $J=H_3(\mathbb{O})$, and define $A^{\circ },J^{\circ }$ as the trace-zero subspaces of $A$ and $J$. Then we have

$$L\left(E_8\right)=\mathrm{Inder}(A)\oplus \left(A^0\otimes J^0\right)\oplus \mathrm{Inder}(J)$$

where

• $\mathrm{Inder}(A)\cong G_2$ has dimension $14$,
• $\mathrm{Inder}(J)\cong F_4$ has dimension $52$,
• $A^0\otimes J^0$ has dimension $7\times 26=182$.

Therefore, the dimension of $L$ is $14+182+52=248$, which perfectly matches the dimension of $E_8$.

The Freudenthal-Tits Magic Square

The construction above is not limited to $E_8$. By varying the composition algebra $A$ and the Jordan algebra $J$ in the Tits construction formula

$$\mathcal{L}(A,J)=\mathrm{Inder}(A)\oplus (A^0\otimes J^0)\oplus \mathrm{Inder}(J),$$

we can generate all the exceptional Lie algebras.

This classification is visualized in the Freudenthal-Tits Magic Square.