Nonassociative rings and algebras Jordan algebras Theorem

Macdonald's theorem on Jordan algebras

Builds on Jordan algebra Special Jordan algebra Field Shirshov–Cohn theorem

Statement

Let $F$ be a field of characteristic different from $2$ . A Jordan polynomial in $x, y, z$ is an expression built from $x$ , $y$ , $z$ by sums, scalar multiples and the Jordan product. It can be evaluated in every Jordan algebra over $F$ . Let $p(x, y, z)$ be a Jordan polynomial in which every term has degree at most $1$ in $z$ . If $p(a, b, c) = 0$ for all elements $a, b, c$ of every special Jordan algebra over $F$ , then $p(a, b, c) = 0$ for all elements $a, b, c$ of every Jordan algebra over $F$ .

Remarks

Proof idea: the identities in question can be rewritten in terms of multiplication operators of $x$ and $y$ applied to $z$ . One studies the algebra generated by these operators in the free Jordan algebra and compares it with the corresponding operator algebra in a free associative algebra. The key point is that two elements always behave as in a special algebra, a fact also expressed by the Shirshov–Cohn theorem.

Example: put $U_a b = 2a(ab) - (aa)b$ . The fundamental formula $U_{U_x y} z = U_x U_y U_x z$ is linear in $z$ , and in a special algebra $A^+$ it reads $(xyx) z (xyx) = x(y(xzx)y)x$ , which is true. By the theorem it holds in every Jordan algebra over $F$ .

The bound on the degree in $z$ is necessary. Glennie found a Jordan identity of degree $8$ in three variables that holds in all special Jordan algebras but fails in the Albert algebra.

Going deeper

For a student

The theorem is the main tool for proving the standard identities of Jordan theory, such as the fundamental formula and its linearizations. Without it, each identity would need a long direct calculation from the Jordan identity. It extends the Shirshov–Cohn theorem, which says that a Jordan algebra generated by two elements is special, so that identities in two variables can be tested in special algebras. Glennie's identities show that one cannot go much further: some identities in three variables hold only in special Jordan algebras.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Field concept
Jordan algebra concept
Special Jordan algebra concept
Shirshov–Cohn theorem statement
Macdonald's theorem on Jordan algebras statement

Tag	`BWYX23HJMF`
URI	`m2://BWYX23HJMF`
Kind	statement, theorem
Writing	W0 stub
MSC	17C05, 17C50
Status	This copy of the site has no build data. Run `m2 status BWYX23HJMF`.
About	Jordan algebra, Special Jordan algebra
Uses	Jordan algebra, Special Jordan algebra, Field, Shirshov–Cohn theorem
Changed	3 November 2026