Cayley-Dickson Construction

References

  • [Baez 2001] “The Octonions”, Baez, John C., Bull. Amer. Math. Soc. 39 (2002), 145-205. (PDF)

  • [Reich 2024] This document. (I haven’t found another reference, yet, for the Theorem stated below, if there is one.)

  • [Schafer 1954] Schafer, R. D. “On the Algebras Formed by the Cayley-Dickson Process.” American Journal of Mathematics 76, no. 2 (1954)

  • [Schafer 1966] “An Introduction to Nonassociative Algebras”, Schafer, R. D., Academic Press, 1966. (or see the 1961 version)

Definitions

The Cayley-Dickson construction begins with a base algebra, \(A\), usually a ring or field, from which another algebra, \(\mathscr{C}(A)\), is derived, with compound elements, identical to those found in the direct product of \(A\) with itself. And, like direct products, addition is defined element-wise, but multiplication is defined similar to complex number multiplication, as described in Definition 1, below. The description is “simplified” because it does not use conjugation anywhere. Conjugation is defined and used in the more general definitions that follow this one.

Definition 1: Cayley-Dickson Algebra (CDA) - Simplified Version (no conjugation)

Let \(A = \langle S, +, \cdot \rangle\) be a ring and let \(T = S \times S\) (i.e., the cross-product of \(S\) with itself).

Then, define the algebra, \(\mathscr{C}(A) \equiv \langle T, \oplus, \odot \rangle\),

where \(\forall (a,b),(c,d) \in T\),

\((a,b) \oplus (c,d) \equiv (a + c, b + d)\)

\((a,b) \odot (c,d) \equiv (a \cdot c - b \cdot d, a \cdot d + b \cdot c)\)

For example, if we apply the Cayley-Dickson construction to the field of real numbers, \(\mathbb{R}\), we obtain the algebra of complex numbers, \(\mathscr{C}(\mathbb{R}) \equiv \mathbb{C}\).

The Cayley-Dickson Algebra (CDA) constructor, \(\mathscr{C}\), can be applied to another CDA, multiple times, as described in Definition 2.

Definition 2: CDA Constructor

Let \(k \in Z^+\) and \(\mathscr{C}^0(A) \equiv A\),

then \(\mathscr{C}^k(A) \equiv \mathscr{C}(\mathscr{C}^{k-1}(A))\),

For example, \(\mathscr{C}^2(\mathbb{R}) \equiv \mathscr{C}(\mathbb{C}) \equiv \mathbb{H}\) is the algebra of quaternions, and \(\mathscr{C}^3(\mathbb{R}) \equiv \mathbb{O}\) is the algebra of octonions.

In many papers on the subject, a generalization of complex multiplication is used, instead of the multiplication described in Definition 1. And, there are multiple, different versions of the generalization.

All of the generalizations begin with a common, recursive definition of conjugation, described in Definition 3.

Definition 3: Conjugation

The conjugate of an algebraic element, \(r\), is denoted by \(\overline{r}\), and defined recursively as follows:

For \(a \in A\), the base algebra, let \(\overline{a} \equiv a\).

Then, let \(k \in Z^+\); \(p,q \in \mathscr{C}^{k-1}(A)\); and \((p,q) \in \mathscr{C}^k(A)\),

and define \(\overline{(p, q)} \equiv (\overline{p}, -q)\)

Three generalizations of multiplication are shown below in Definition 4 (versions A, B, & C):

Definition 4: CDA Multiplication

Let \(k \in Z^+\); \(a,b,c,d,\mu \in \mathscr{C}^{k-1}(A)\); and \(\mu \neq 0\),

then define multiplication for \((a,b), (c,d) \in \mathscr{C}^k(A)\) according to one of the following definitions:

[Shafer 1966] Version 4-A:

\((a, b) \odot (c, d) = (a \cdot c + \mu \cdot d \cdot \overline{b}, \overline{a} \cdot d + c \cdot b)\)

[Shafer 1954] Version 4-B:

\((a, b) \odot (c, d) = (a \cdot c + \mu \cdot \overline{d} \cdot b, d \cdot a + b \cdot \overline{c})\)

[Baez 2001] Version 4-C:

\((a,b) \odot (c,d) \equiv (a \cdot c - d \cdot \overline{b}, \overline{a} \cdot d + c \cdot b)\)

[Reich 2024] Definition 1, again (for comparison)

\((a,b) \odot (c,d) \equiv (a \cdot c - b \cdot d, a \cdot d + b \cdot c)\)

Note that, if the base (“starting”) algebra, \(A\), is a field, then multiplication is commutative and every element is it’s own conjugate. And, if \(\mu = -1\), where \(-1\) is the additive inverse of \(A\)’s multiplicative identity element, then all of the multiplication versions above produce the same algebra, \(\mathscr{C}(A)\). But, when we start applying CDA constructions on top of other CDA constructions, the different versions diverge. And, not only that, even if we stick to a single multiplication version, each subsequent application of the CDA construction process results in increasingly “unstructured” algebras. As [Baez 2001] writes, “…each time we apply the construction, our algebra gets a bit worse. First we lose the fact that every element is its own conjugate, then we lose commutativity, then we lose associativity, and finally we lose the division algebra property.”

Finite Cayley-Dickson Algebras

The Cayley-Dickson constructor, \(\mathscr{C}\), defined above, can also be applied to finite rings or fields.

An interesting result regarding the application of the Cayley-Dickson construction to a finite field is stated in the Theorem below.

Theorem [Reich 2024]

Let \(F_n\) be a finite field of order \(n\), where \(n \in \mathbb{Z}^+\) and \(n > 1\) is a Gaussian prime (e.g., 3, 7, 11, 19, 23, 31, 43, …),

then \(\mathscr{C}(F_n)\) will be a finite field of order \(n^2\),

otherwise \(\mathscr{C}(F_n)\) will be a finite ring.

Examples

Since each application of the Cayley-Dickson construction doubles the number of elements of its base algebra, repeated applications of the construction can result in much larger algebras, so the examples in this section will be restricted to small base rings & fields.

import finite_algebras as alg

The finite_algebras function, generate_algebra_mod_n, will generate a ring or field based on integer addition and multiplication modulo n. If n is prime, then result will be a field, otherwise it will be a ring.

F3 = alg.generate_algebra_mod_n(3)
F3.about()

** Field **
Name: F3
Instance ID: 4969282064
Description: Autogenerated Field of integers mod 3
Order: 3
Identity: '0'
Commutative? Yes
Cyclic?: Yes
Generators: ['2', '1']
Elements:
   Index   Name   Inverse  Order
      0     '0'     '0'       1
      1     '1'     '2'       3
      2     '2'     '1'       3
Cayley Table (showing indices):
[[0, 1, 2], [1, 2, 0], [2, 0, 1]]
Mult. Identity: '1'
Mult. Commutative? Yes
Zero Divisors: None
Multiplicative Cayley Table (showing indices):
[[0, 0, 0], [0, 1, 2], [0, 2, 1]]

Since 3 is a Gaussian prime, the following field of integers modulo 3 will give us an opportunity to test the theorem stated above.

CDAs of F3 (4 versions)

The method make_cayley_dickson_algebra applies the Cayley-Dickson (CD) construction to a Ring or Field.

It’s help documentation is shown below.

help(F3.make_cayley_dickson_algebra)

Help on method make_cayley_dickson_algebra in module finite_algebras:

make_cayley_dickson_algebra(mu=None, version=1) method of finite_algebras.Field instance
    Constructs the Cayley-Dickson algebra using this Ring or Field.

    Several different versions of multiplication are supported:
    version=1: (DEFAULT) No mu & no conjugation are used
    version=2: Definition in Schafer, 1966
    version=3: Definition in Schafer, 1954
    version=4: Definition in Baez, 2001.

    See the documentation on readthedocs for more information regarding versions.

    Versions 2 & 3 require a value for mu. If mu is None (the default), then mu
    will be automatically set to be the additive inverse of the Ring's
    multiplicative identity element (i.e., "-1") if it exists. If it does not
    exist, then an exception will be raised.

The following computation creates 4 versions if \(\mathscr{C}(F_3)\) and prints out the name stored internally in each CDA.

F3cda24 = F3.make_cayley_dickson_algebra(version=1); print(F3cda24.name)
F3cda66 = F3.make_cayley_dickson_algebra(version=2); print(F3cda66.name)
F3cda54 = F3.make_cayley_dickson_algebra(version=3); print(F3cda54.name)
F3cda01 = F3.make_cayley_dickson_algebra(version=4); print(F3cda01.name)

F3_CDA_2024
F3_CDA_1966
F3_CDA_1954
F3_CDA_2001

Here’s the [Schafer 1966] version (i.e., version 2). As we’ll see below, all four versions are the same. Also, note that \(\mathscr{C}(F_3)\) is a field.

F3cda66.about()

** Field **
Name: F3_CDA_1966
Instance ID: 4970233936
Description: Cayley-Dickson algebra based on F3, where mu = 2, Schafer 1966 version.
Order: 9
Identity: '0:0'
Commutative? Yes
Cyclic?: Yes
Generators: ['0:1', '1:2', '0:2', '2:2', '1:1', '2:1']
Elements:
   Index   Name   Inverse  Order
      0   '0:0'   '0:0'       1
      1   '0:1'   '0:2'       3
      2   '0:2'   '0:1'       3
      3   '1:0'   '2:0'       3
      4   '1:1'   '2:2'       3
      5   '1:2'   '2:1'       3
      6   '2:0'   '1:0'       3
      7   '2:1'   '1:2'       3
      8   '2:2'   '1:1'       3
Cayley Table (showing indices):
[[0, 1, 2, 3, 4, 5, 6, 7, 8],
 [1, 2, 0, 4, 5, 3, 7, 8, 6],
 [2, 0, 1, 5, 3, 4, 8, 6, 7],
 [3, 4, 5, 6, 7, 8, 0, 1, 2],
 [4, 5, 3, 7, 8, 6, 1, 2, 0],
 [5, 3, 4, 8, 6, 7, 2, 0, 1],
 [6, 7, 8, 0, 1, 2, 3, 4, 5],
 [7, 8, 6, 1, 2, 0, 4, 5, 3],
 [8, 6, 7, 2, 0, 1, 5, 3, 4]]
Mult. Identity: '1:0'
Mult. Commutative? Yes
Zero Divisors: None
Multiplicative Cayley Table (showing indices):
[[0, 0, 0, 0, 0, 0, 0, 0, 0],
 [0, 6, 3, 1, 7, 4, 2, 8, 5],
 [0, 3, 6, 2, 5, 8, 1, 4, 7],
 [0, 1, 2, 3, 4, 5, 6, 7, 8],
 [0, 7, 5, 4, 2, 6, 8, 3, 1],
 [0, 4, 8, 5, 6, 1, 7, 2, 3],
 [0, 2, 1, 6, 8, 7, 3, 5, 4],
 [0, 8, 4, 7, 3, 2, 5, 1, 6],
 [0, 5, 7, 8, 1, 3, 4, 6, 2]]

from itertools import combinations

cdas = [F3cda24, F3cda66, F3cda54, F3cda01]

for cda in cdas:
    print(f"{cda.name} is a {cda.__class__.__name__}")

print("")

for pair in list(combinations(cdas, 2)):
    print(f"{pair[0].name} == {pair[1].name} ? {alg.yes_or_no(pair[0] == pair[1])}")

F3_CDA_2024 is a Field
F3_CDA_1966 is a Field
F3_CDA_1954 is a Field
F3_CDA_2001 is a Field

F3_CDA_2024 == F3_CDA_1966 ? Yes
F3_CDA_2024 == F3_CDA_1954 ? Yes
F3_CDA_2024 == F3_CDA_2001 ? Yes
F3_CDA_1966 == F3_CDA_1954 ? Yes
F3_CDA_1966 == F3_CDA_2001 ? Yes
F3_CDA_1954 == F3_CDA_2001 ? Yes

CDAs of F5 (4 versions)

This section goes through the same calculations as above, except that \(F_5\) is used, instead of \(F_3\).

This time, all four versions if the CDA, \(\mathscr{C}(F_5)\), will be the same ring.

F5 = alg.generate_algebra_mod_n(5)
F5.about()

** Field **
Name: F5
Instance ID: 4969990480
Description: Autogenerated Field of integers mod 5
Order: 5
Identity: '0'
Commutative? Yes
Cyclic?: Yes
Generators: ['2', '3', '4', '1']
Elements:
   Index   Name   Inverse  Order
      0     '0'     '0'       1
      1     '1'     '4'       5
      2     '2'     '3'       5
      3     '3'     '2'       5
      4     '4'     '1'       5
Cayley Table (showing indices):
[[0, 1, 2, 3, 4],
 [1, 2, 3, 4, 0],
 [2, 3, 4, 0, 1],
 [3, 4, 0, 1, 2],
 [4, 0, 1, 2, 3]]
Mult. Identity: '1'
Mult. Commutative? Yes
Zero Divisors: None
Multiplicative Cayley Table (showing indices):
[[0, 0, 0, 0, 0],
 [0, 1, 2, 3, 4],
 [0, 2, 4, 1, 3],
 [0, 3, 1, 4, 2],
 [0, 4, 3, 2, 1]]

F5cda24 = F5.make_cayley_dickson_algebra(version=1); print(F5cda24.name)
F5cda66 = F5.make_cayley_dickson_algebra(version=2); print(F5cda66.name)
F5cda54 = F5.make_cayley_dickson_algebra(version=3); print(F5cda54.name)
F5cda01 = F5.make_cayley_dickson_algebra(version=4); print(F5cda01.name)

F5_CDA_2024
F5_CDA_1966
F5_CDA_1954
F5_CDA_2001

Here’s a representative example, the [Schafer 1966] version:

F5cda66.about()

** Ring **
Name: F5_CDA_1966
Instance ID: 4969316048
Description: Cayley-Dickson algebra based on F5, where mu = 4, Schafer 1966 version.
Order: 25
Identity: '0:0'
Commutative? Yes
Cyclic?: Yes
Generators: ['0:1', '4:4', '3:2', '0:3', '2:3', '1:4', '3:3', '4:1', '1:1', '0:4', '0:2', '2:2']
Elements:
   Index   Name   Inverse  Order
      0   '0:0'   '0:0'       1
      1   '0:1'   '0:4'       5
      2   '0:2'   '0:3'       5
      3   '0:3'   '0:2'       5
      4   '0:4'   '0:1'       5
      5   '1:0'   '4:0'       5
      6   '1:1'   '4:4'       5
      7   '1:2'   '4:3'       5
      8   '1:3'   '4:2'       5
      9   '1:4'   '4:1'       5
     10   '2:0'   '3:0'       5
     11   '2:1'   '3:4'       5
     12   '2:2'   '3:3'       5
     13   '2:3'   '3:2'       5
     14   '2:4'   '3:1'       5
     15   '3:0'   '2:0'       5
     16   '3:1'   '2:4'       5
     17   '3:2'   '2:3'       5
     18   '3:3'   '2:2'       5
     19   '3:4'   '2:1'       5
     20   '4:0'   '1:0'       5
     21   '4:1'   '1:4'       5
     22   '4:2'   '1:3'       5
     23   '4:3'   '1:2'       5
     24   '4:4'   '1:1'       5
Ring order is 25 > 12, so no table is printed.
Mult. Identity: '1:0'
Mult. Commutative? Yes
Zero Divisors: ['1:2', '1:3', '2:1', '2:4', '3:1', '3:4', '4:2', '4:3']
Ring order is 25 > 12, so the mult. table is not printed.

cdas = [F5cda24, F5cda66, F5cda54, F5cda01]

for cda in cdas:
    print(f"{cda.name} is a {cda.__class__.__name__}")

print("")

for pair in list(combinations(cdas, 2)):
    print(f"{pair[0].name} == {pair[1].name} ? {alg.yes_or_no(pair[0] == pair[1])}")

F5_CDA_2024 is a Ring
F5_CDA_1966 is a Ring
F5_CDA_1954 is a Ring
F5_CDA_2001 is a Ring

F5_CDA_2024 == F5_CDA_1966 ? Yes
F5_CDA_2024 == F5_CDA_1954 ? Yes
F5_CDA_2024 == F5_CDA_2001 ? Yes
F5_CDA_1966 == F5_CDA_1954 ? Yes
F5_CDA_1966 == F5_CDA_2001 ? Yes
F5_CDA_1954 == F5_CDA_2001 ? Yes

And, here’s the multiplicative portion of this algebra. It’s a monoid.

F5cda66.extract_multiplicative_algebra().about()

** Monoid **
Name: F5_CDA_1966.Mult
Instance ID: 4970332960
Description: Multiplicative-only portion of F5_CDA_1966
Order: 25
Identity: 1:0
Associative? Yes
Commutative? Yes
Cyclic?: No
Elements: ('0:0', '0:1', '0:2', '0:3', '0:4', '1:0', '1:1', '1:2', '1:3', '1:4', '2:0', '2:1', '2:2', '2:3', '2:4', '3:0', '3:1', '3:2', '3:3', '3:4', '4:0', '4:1', '4:2', '4:3', '4:4')
Has Inverses? No
Monoid order is 25 > 12, so the table is not output.

Quads (CDA of a CDA)

Four CDAs, each one a \(\mathscr{C}^2(F_3)\), are generated below, each one using a different version of multiplication, per the discussion above.

%%time

F3quad24 = F3cda24.make_cayley_dickson_algebra(version=1)
F3quad66 = F3cda66.make_cayley_dickson_algebra(version=2)
F3quad54 = F3cda54.make_cayley_dickson_algebra(version=3)
F3quad01 = F3cda01.make_cayley_dickson_algebra(version=4)

CPU times: user 3.19 s, sys: 12.5 ms, total: 3.2 s
Wall time: 3.24 s

The computation below shows that the only versions of \(\mathscr{C}^2(F_3)\) that are equal to each other are [Schafer 1966] and [Baez 2001].

from itertools import combinations

quad_cdas = [F3quad24, F3quad66, F3quad54, F3quad01]

for pair in list(combinations(quad_cdas, 2)):
    print(f"{pair[0].name} == {pair[1].name} ? {alg.yes_or_no(pair[0] == pair[1])}")

F3_CDA_2024_CDA_2024 == F3_CDA_1966_CDA_1966 ? No
F3_CDA_2024_CDA_2024 == F3_CDA_1954_CDA_1954 ? No
F3_CDA_2024_CDA_2024 == F3_CDA_2001_CDA_2001 ? No
F3_CDA_1966_CDA_1966 == F3_CDA_1954_CDA_1954 ? No
F3_CDA_1966_CDA_1966 == F3_CDA_2001_CDA_2001 ? Yes
F3_CDA_1954_CDA_1954 == F3_CDA_2001_CDA_2001 ? No

F3quad66.about(show_conjugates=False)

** Ring **
Name: F3_CDA_1966_CDA_1966
Instance ID: 4403910640
Description: Cayley-Dickson algebra based on F3_CDA_1966, where mu = 2:0, Schafer 1966 version.
Order: 81
Identity: '0:0:0:0'
Commutative? Yes
Cyclic?: No
Elements:
   Index   Name   Inverse  Order
      0 '0:0:0:0' '0:0:0:0'       1
      1 '0:0:0:1' '0:0:0:2'       3
      2 '0:0:0:2' '0:0:0:1'       3
      3 '0:0:1:0' '0:0:2:0'       3
      4 '0:0:1:1' '0:0:2:2'       3
      5 '0:0:1:2' '0:0:2:1'       3
      6 '0:0:2:0' '0:0:1:0'       3
      7 '0:0:2:1' '0:0:1:2'       3
      8 '0:0:2:2' '0:0:1:1'       3
      9 '0:1:0:0' '0:2:0:0'       3
     10 '0:1:0:1' '0:2:0:2'       3
     11 '0:1:0:2' '0:2:0:1'       3
     12 '0:1:1:0' '0:2:2:0'       3
     13 '0:1:1:1' '0:2:2:2'       3
     14 '0:1:1:2' '0:2:2:1'       3
     15 '0:1:2:0' '0:2:1:0'       3
     16 '0:1:2:1' '0:2:1:2'       3
     17 '0:1:2:2' '0:2:1:1'       3
     18 '0:2:0:0' '0:1:0:0'       3
     19 '0:2:0:1' '0:1:0:2'       3
     20 '0:2:0:2' '0:1:0:1'       3
     21 '0:2:1:0' '0:1:2:0'       3
     22 '0:2:1:1' '0:1:2:2'       3
     23 '0:2:1:2' '0:1:2:1'       3
     24 '0:2:2:0' '0:1:1:0'       3
     25 '0:2:2:1' '0:1:1:2'       3
     26 '0:2:2:2' '0:1:1:1'       3
     27 '1:0:0:0' '2:0:0:0'       3
     28 '1:0:0:1' '2:0:0:2'       3
     29 '1:0:0:2' '2:0:0:1'       3
     30 '1:0:1:0' '2:0:2:0'       3
     31 '1:0:1:1' '2:0:2:2'       3
     32 '1:0:1:2' '2:0:2:1'       3
     33 '1:0:2:0' '2:0:1:0'       3
     34 '1:0:2:1' '2:0:1:2'       3
     35 '1:0:2:2' '2:0:1:1'       3
     36 '1:1:0:0' '2:2:0:0'       3
     37 '1:1:0:1' '2:2:0:2'       3
     38 '1:1:0:2' '2:2:0:1'       3
     39 '1:1:1:0' '2:2:2:0'       3
     40 '1:1:1:1' '2:2:2:2'       3
     41 '1:1:1:2' '2:2:2:1'       3
     42 '1:1:2:0' '2:2:1:0'       3
     43 '1:1:2:1' '2:2:1:2'       3
     44 '1:1:2:2' '2:2:1:1'       3
     45 '1:2:0:0' '2:1:0:0'       3
     46 '1:2:0:1' '2:1:0:2'       3
     47 '1:2:0:2' '2:1:0:1'       3
     48 '1:2:1:0' '2:1:2:0'       3
     49 '1:2:1:1' '2:1:2:2'       3
     50 '1:2:1:2' '2:1:2:1'       3
     51 '1:2:2:0' '2:1:1:0'       3
     52 '1:2:2:1' '2:1:1:2'       3
     53 '1:2:2:2' '2:1:1:1'       3
     54 '2:0:0:0' '1:0:0:0'       3
     55 '2:0:0:1' '1:0:0:2'       3
     56 '2:0:0:2' '1:0:0:1'       3
     57 '2:0:1:0' '1:0:2:0'       3
     58 '2:0:1:1' '1:0:2:2'       3
     59 '2:0:1:2' '1:0:2:1'       3
     60 '2:0:2:0' '1:0:1:0'       3
     61 '2:0:2:1' '1:0:1:2'       3
     62 '2:0:2:2' '1:0:1:1'       3
     63 '2:1:0:0' '1:2:0:0'       3
     64 '2:1:0:1' '1:2:0:2'       3
     65 '2:1:0:2' '1:2:0:1'       3
     66 '2:1:1:0' '1:2:2:0'       3
     67 '2:1:1:1' '1:2:2:2'       3
     68 '2:1:1:2' '1:2:2:1'       3
     69 '2:1:2:0' '1:2:1:0'       3
     70 '2:1:2:1' '1:2:1:2'       3
     71 '2:1:2:2' '1:2:1:1'       3
     72 '2:2:0:0' '1:1:0:0'       3
     73 '2:2:0:1' '1:1:0:2'       3
     74 '2:2:0:2' '1:1:0:1'       3
     75 '2:2:1:0' '1:1:2:0'       3
     76 '2:2:1:1' '1:1:2:2'       3
     77 '2:2:1:2' '1:1:2:1'       3
     78 '2:2:2:0' '1:1:1:0'       3
     79 '2:2:2:1' '1:1:1:2'       3
     80 '2:2:2:2' '1:1:1:1'       3
Ring order is 81 > 12, so no table is printed.
Mult. Identity: '1:0:0:0'
Mult. Commutative? No
Zero Divisors: ['0:1:1:1', '0:1:1:2', '0:1:2:1', '0:1:2:2', '0:2:1:1', '0:2:1:2', '0:2:2:1', '0:2:2:2', '1:0:1:1', '1:0:1:2', '1:0:2:1', '1:0:2:2', '1:1:0:1', '1:1:0:2', '1:1:1:0', '1:1:2:0', '1:2:0:1', '1:2:0:2', '1:2:1:0', '1:2:2:0', '2:0:1:1', '2:0:1:2', '2:0:2:1', '2:0:2:2', '2:1:0:1', '2:1:0:2', '2:1:1:0', '2:1:2:0', '2:2:0:1', '2:2:0:2', '2:2:1:0', '2:2:2:0']
Ring order is 81 > 12, so the mult. table is not printed.

Zero Divisors

The next two calculations show that the version of \(\mathscr{C}^2(F_3)\) that uses [Reich 2024] has 16 zero divisors, while the other three versions each have 32 zero divisors.

And, the zero divisors of the [Schafer 1966], [Schafer 1954], and [Baez 2001] versions are all the same, while the zero divisors of the [Reich 2024] version have nothing in common with them.

for quad in quad_cdas:
    print(f"{quad.name} has {len(quad.zero_divisors())} zero divisors")

F3_CDA_2024_CDA_2024 has 16 zero divisors
F3_CDA_1966_CDA_1966 has 32 zero divisors
F3_CDA_1954_CDA_1954 has 32 zero divisors
F3_CDA_2001_CDA_2001 has 32 zero divisors

for pair in list(combinations(quad_cdas, 2)):
    zd0 = set(pair[0].zero_divisors())
    zd1 = set(pair[1].zero_divisors())
    zd_in_common = zd0.intersection(zd1)
    print(f"{pair[0].name} & {pair[1].name} have {len(zd_in_common)} zero divisors in common.")

F3_CDA_2024_CDA_2024 & F3_CDA_1966_CDA_1966 have 0 zero divisors in common.
F3_CDA_2024_CDA_2024 & F3_CDA_1954_CDA_1954 have 0 zero divisors in common.
F3_CDA_2024_CDA_2024 & F3_CDA_2001_CDA_2001 have 0 zero divisors in common.
F3_CDA_1966_CDA_1966 & F3_CDA_1954_CDA_1954 have 32 zero divisors in common.
F3_CDA_1966_CDA_1966 & F3_CDA_2001_CDA_2001 have 32 zero divisors in common.
F3_CDA_1954_CDA_1954 & F3_CDA_2001_CDA_2001 have 32 zero divisors in common.

sorted(F3quad24.zero_divisors())

['0:1:1:0',
 '0:1:2:0',
 '0:2:1:0',
 '0:2:2:0',
 '1:0:0:1',
 '1:0:0:2',
 '1:1:1:2',
 '1:1:2:1',
 '1:2:1:1',
 '1:2:2:2',
 '2:0:0:1',
 '2:0:0:2',
 '2:1:1:1',
 '2:1:2:2',
 '2:2:1:2',
 '2:2:2:1']

sorted(F3quad66.zero_divisors())

['0:1:1:1',
 '0:1:1:2',
 '0:1:2:1',
 '0:1:2:2',
 '0:2:1:1',
 '0:2:1:2',
 '0:2:2:1',
 '0:2:2:2',
 '1:0:1:1',
 '1:0:1:2',
 '1:0:2:1',
 '1:0:2:2',
 '1:1:0:1',
 '1:1:0:2',
 '1:1:1:0',
 '1:1:2:0',
 '1:2:0:1',
 '1:2:0:2',
 '1:2:1:0',
 '1:2:2:0',
 '2:0:1:1',
 '2:0:1:2',
 '2:0:2:1',
 '2:0:2:2',
 '2:1:0:1',
 '2:1:0:2',
 '2:1:1:0',
 '2:1:2:0',
 '2:2:0:1',
 '2:2:0:2',
 '2:2:1:0',
 '2:2:2:0']

Something to Note: The computation below, shows that all 32 of the zero divisors of F3_CDA_1966_CDA_1966, F3_CDA_1954_CDA_1954, and F3_CDA_2001_CDA_2001 have exactly one ‘0’ as a component. Basically, an element is a zero divisor *if and only if it has exactly one ‘0’ component.

There does not seem to be an obvious pattern to the 16 zero divisors of F3_CDA_2024_CDA_2024.

from operator import countOf

algebra = F3quad66

count = 0
for elem in algebra.elements:
    num = countOf(elem.split(':'), '0')
    if num == 1:
        count += 1

print(count)

32

Regarding the Theorem

%%time

n = 24
#n = 7
vers = 2

for i in range(2,n + 1):
    fi = alg.generate_algebra_mod_n(i)
    fi_cda = fi.make_cayley_dickson_algebra(version=vers)
    class_name = fi_cda.__class__.__name__
    if class_name == 'Field':
        print(f"{class_name}: {fi_cda.name} <-- Gaussian prime: {i}")
    else:
        print(f"{class_name}: {fi_cda.name}")

print()

Ring: F2_CDA_1966
Field: F3_CDA_1966 <-- Gaussian prime: 3
Ring: R4_CDA_1966
Ring: F5_CDA_1966
Ring: R6_CDA_1966
Field: F7_CDA_1966 <-- Gaussian prime: 7
Ring: R8_CDA_1966
Ring: R9_CDA_1966
Ring: R10_CDA_1966
Field: F11_CDA_1966 <-- Gaussian prime: 11
Ring: R12_CDA_1966
Ring: F13_CDA_1966
Ring: R14_CDA_1966
Ring: R15_CDA_1966
Ring: R16_CDA_1966
Ring: F17_CDA_1966
Ring: R18_CDA_1966
Field: F19_CDA_1966 <-- Gaussian prime: 19
Ring: R20_CDA_1966
Ring: R21_CDA_1966
Ring: R22_CDA_1966
Field: F23_CDA_1966 <-- Gaussian prime: 23
Ring: R24_CDA_1966

CPU times: user 19min 48s, sys: 2.34 s, total: 19min 50s
Wall time: 20min 28s

Additional Information