The Eight Cayley–Dickson Doubling Products

Abstract

The purpose of this paper is to identify all eight of the basic Cayley–Dickson doubling products. A Cayley–Dickson algebra AN+1{\mathbb{A}_{N+1}} of dimension 2N+1{2^{N+1}} consists of all ordered pairs of elements of a Cayley–Dickson algebra AN{\mathbb{A}_{N}} of dimension 2N{2^N} where the product (a,b)(c,d){(a, b)(c, d)} of elements of AN+1{\mathbb{A}_{N+1}} is defined in terms of a pair of second degree binomials (f(a,b,c,d),g(a,b,c,d)){(f(a, b, c, d), g(a, b, c,d))} satisfying certain properties. The polynomial pair(f,g){(f, g)} is called a ‘doubling product.’ While A0{\mathbb{A}_{0}} may denote any ring, here it is taken to be the set R{\mathbb{R}} of real numbers. The binomials f{f} and g{g} should be devised such that A1=C{\mathbb{A}_{1} = \mathbb{C}} the complex numbers, A2=H{\mathbb{A}_{2} = \mathbb{H}} the quaternions, and A3=O{\mathbb{A}_{3} = \mathbb{O}} the octonions. Historically, various researchers have used different yet equivalent doubling products.

Full text

The eight Cayley-Dickson doubling products
John W. Bales
Abstract. The purpose of this paper is to identify all eight of the basic
Cayley-Dickson doubling products. A Cayley-Dickson algebra AN+1 of
dimension 2N+1 consists of all ordered pairs of elements of a Cayley-
Dickson algebra ANof dimension 2Nwhere the product (a, b)(c, d) of
elements of AN+1 is defined in terms of a pair of second degree binomials
(f(a, b, c, d), g(a, b, c, d)) satisfying certain properties. The polynomial
pair(f, g ) is called a ‘doubling product.’ While A0may denote any ring,
here it is taken to be the set Rof real numbers. The binomials fand g
should be devised such that A1=Cthe complex numbers, A2=Hthe
quaternions, and A3=Othe octonions. Historically, various researchers
have used different yet equivalent doubling products.
Disclaimer: The final official version of this paper was published at
Springer and is available on their website at the following link.
https://link.springer.com/article/10.1007/s00006-015-0638-6
Mathematics Subject Classification (2010). 16S99,16W99.
Keywords. Cayley-Dickson algebra, doubling product, twisted group al-
gebra, quaternions, octonions, Fano plane.

2 John W. Bales
1. Introduction
Although it is recognized that there are several possible Cayley-Dickson dou-
bling products, past researchers have restricted themselves to only two of
them. Furthermore, the development of the basis vectors has proceeded his-
torically in a fashion which obscures the high periodicity of the structure
constants associated with the products of the Cayley-Dickson basis vectors.
The purpose of this paper is to catalog all possible variants of the Cayley-
Dickson doubling product and to suggest an alternate way to number the
basis vectors. The alternate numbering method has been used in the past,
for example in Shafer’s 1954 paper [14], but has fallen out of favor. By iden-
tifying the ordered pair of two sequences as the ‘shuffling’ of the two se-
quences the author demonstrates how this leads naturally to the numbering
used by Shafer and others and how it illustrates the periodicity of the struc-
ture constants and corresponding ‘twist’ on the group product underlying
the product of the basis vectors. As a result of cataloging the eight different
Cayley-Dickson doubling products the author has identified one in particular
(a, b)(c, d) = (ac −b∗d, da∗+bc) having an interesting twist map. To the
author’s knowledge this particular doubling product has never been investi-
gated.
2. Defining properties
We shall take the following as necessary properties of a Cayley-Dickson alge-
bra AN:
1. For each Nand each x∈AN,
1·x=x·1 = x(2.1)
2. There shall be a norm and an involution ∗such that for each Nand
each x, y ∈AN,
x+x∗∈A0=R(2.2)
xx∗=x∗x=kxk2(2.3)
(xy)∗=y∗x∗(2.4)
3. There shall be an infinite sequence e0, e1, e2,··· of unit vectors such
that
(a) For each N≥0 if p < 2Nthen epis a basis vector for AN
(b) The set {±ep|0≤p < 2N}is a group under the product on AN.
(c) e0= 1 and if p > 0 then
e2
p=−e0(2.5)
(d) If 0 6=p6=q6= 0 then the anti-symmetric property holds:
epeq+eqep= 0 (2.6)

The eight Cayley-Dickson doubling products 3
(e) If 0 6=p6=q6= 0 then there is an r6= 0 such that either epeq=er
or epeq=−erand the quaternion property holds:
epeq=erimplies eqer=ep(2.7)
3. The basis vectors
The entire approach taken in this paper is highly dependent upon the manner
in which the basis vectors are chosen. They are deliberately chosen to be the
basis vectors of the Hilbert space ℓ2of square-summable sequences. That is,
e0= 1,0,0,0,···
e1= 0,1,0,0,···
e2= 0,0,1,0,···
.
.
.
In order to relate these basis vectors to the idea of ordered pairs, we identify
a real number rwith the sequence
r=r, 0,0,0,··· (3.1)
and the ordered pair of two sequences
x=x0, x1, x2,···
y=y0, y1, y2,···
with the ‘shuffle’ of those two sequences
(x, y) = x0, y0, x1, y1, x2, y2,···
This ‘shuffle basis’ immediately leads to two results: an inductive devel-
opment of the basis vectors and the proper definition of the involution.
These basis vectors defined inductively as ordered pairs are
e0= 1 (3.2)
e2p= (ep,0) (3.3)
e2p+1 = (0, ep) (3.4)
Furthermore, since x+x∗∈Ris a requirement it is sufficient that
x∗=x0,−x1,−x2,··· (3.5)
with the result that
(x, y)∗=x0,−y0,−x1,−y1,···
which, expressed in terms of ordered pairs is
(x, y)∗= (x∗,−y) (3.6)
which is the traditional involution for Cayley-Dickson algebras. This also
guarantees by induction that for each AN

4 John W. Bales
x∗∗ =x(3.7)
4. Devising an adequate product
It is required that 1 ·x=x·1 = x.
Let (a, b)(c, d) = (f(a, b, c, d), g(a, b, c, d)).
For it to be the case that (a, b)(1,0) = (a, b) and (1,0)(c, d) = (c, d)
it is sufficient that fcontain one of the terms ac or ca plus or minus some
product of bor b∗with dor d∗.
Furthermore, it is sufficient that gcontain some product of aor a∗with
dplus a product of cor c∗with b.
It is required that xx∗=x∗x=kxk. Thus
(a, b)(a∗,−b) = kak2+kbk2,0and (c∗,−d)(c, d) = kck2+kdk2,0.
Thus fmust contain a product of aand cor a∗and c∗minus a product
of either dand b∗or a product of band d∗. Combined with the previous result
it suffices for fto equal one of the following binomials:
f0(a, b, c, d) = ca −b∗d
f1(a, b, c, d) = ca −db∗
f2(a, b, c, d) = ac −b∗d
f3(a, b, c, d) = ac −db∗
f4(a, b, c, d) = ca −bd∗
f5(a, b, c, d) = ca −d∗b
f6(a, b, c, d) = ac −bd∗
f7(a, b, c, d) = ac −d∗b
We already know that gmust contain one of the following terms: ad,
da,a∗dor da∗. In order for gto be zero when an ordered pair is multiplied
by its conjugate it must be the case that, for each of these four options g
must contain, respectively, the terms c∗b,bc∗,cb and bc.
Therefore gmust be one of the following four binomials:
g0(a, b, c, d) = da∗+bc
g1(a, b, c, d) = a∗d+cb
g2(a, b, c, d) = ad +c∗b
g3(a, b, c, d) = da +bc∗

The eight Cayley-Dickson doubling products 5
This results in a combination of 32 possible Cayley-Dickson doubling
products. And indeed all 32 can be shown to satisfy properties (2.1) through
(2.6) above. It is the quaternion property (2.7) which reduces the number
from thirty-two to eight.
5. Twisted Group Algebra
The set W={0,1,2,3,· · · } of whole numbers is a group under the bit-wise
‘exclusive or’ (XOR) operation on their binary representations, with group
identity 0. For example, 27 ⊕14 = 11011b⊕01110b= 10101b= 21. Rather
than representing the group operation as addition, we will use juxtaposition.
Thus for p, q ∈Wwe will denote p⊕qas simply pq. This group operation
on W2is pertinent since for p, q ∈Wand each of the 32 doubling products
(fi, gj) there is an ωij (p, q)∈ {−1,1}such that
epeq=ωij (p, q)epq (5.1)
The map ωij :W×W→ {−1,1}is called a twist on the group Wmaking
each of the algebras resulting from the 32 doubling products a twisted group
algebra.[9] [6] [13].
6. Interior points of W2and their successors
Property (2.7) is a property of the interior points of W2. The interior points
of W2are the points (p, q)∈W2such that 0 6=p6=q6= 0. Every point of
W2has four successor points (2p, 2q),(2p, 2q+ 1),(2p+ 1,2q)(2p+ 1,2q+ 1).
Interior points of W2which are not a successor point of an interior point are
initial interior points. To establish a general property of the interior points
of W2by induction one must first establish that the property holds for the
initial interior points (the basis step). The inductive step consists of showing
that if the property holds for an interior point (p, q) then the property holds
for the four successors of (p, q).
We will find that for the initial interior points property (2.7) holds for
only half of the 32 possible doubling products. Then we will find that only
8 of those 16 doubling products will satisfy property (2.7) for the remaining
interior points of W2.
Before beginning with the induction, let us re-express property (2.7)
with property (5.1) in mind.
If 0 6=p6=q6= 0 and if epeq=epq then eqepq =epand epq ep=eq.(6.1)
We will adopt the notation (p, q, r) to mean that epeq=er. Since epeq+
eqep= 0 for 0 6=p6=q, one but not both of (p, q, pq) or (q, p, pq ) must be the
case.

6 John W. Bales
Table 7.1. First Elimination of Doubling Products
f0f1f2f3f4f5f6f7
g0Q⊤Q⊤Q⊤Q⊤Q⊤Q⊤Q⊤Q⊤
Q⊤Q⊤Q⊤Q⊤QQQQ
g1Q⊤Q⊤Q⊤Q⊤Q⊤Q⊤Q⊤Q⊤
Q⊤Q⊤Q⊤Q⊤QQQQ
g2Q Q Q Q Q Q Q Q
Q⊤Q⊤Q⊤Q⊤QQQQ
g3Q Q Q Q Q Q Q Q
Q⊤Q⊤Q⊤Q⊤QQQQ
7. The basis step of the inductive proof of property (6.1)
First we will determine which of the 32 doubling products satisfy property
(6.1) for basis vector products epeqwhere (p, q ) is an initial interior point
of W2. The initial interior points are precisely those interior points (p, q) for
which either p= 1 or q= 1 or for which pand qdiffer by exactly 1. Put
another way, the initial interior points of W2consist of the ordered pairs of
the forms (2s, 1),(1,2s),(2s+ 1,1),(1,2s+ 1),(2s, 2s+ 1),(2s+ 1,2s) where
s > 0.
For each interior point of W2we must have either one or the other but
not both of the following quaternion properties:
(2s, 1,2s+ 1) and (1,2s+ 1,2s) and (2s+ 1,2s, 1) (Q)
(1,2s, 2s+ 1) and (2s, 2s+ 1,1) and (2s+ 1,1,2s) (Q⊤)
Consider the following
e2se1= (es,0)(0,1) = ((0, es) = e2s+1 for g2, g3with all fi
(0,−es) = −e2s+1 for g0, g1with all fi
(7.1)
e1e2s+1 = (0,1)(0, es) = ((es,0) = e2sfor f4, f5, f6, f7with all gj
(−es,0) = −e2sfor f0, f1, f2, f3with all gj
(7.2)
e2s+1e2s= (0, es)(es,0) = ((0,1) = e1for g2, g3with all fi
(0,−1) = −e1for g0, g1with all fi
(7.3)
To make sense of these results we summarize in table 7.1 which of the
32 doubling products satisfy a condition from either option Qor Q⊤. Any
doubling product (shown in gray) which satisfies a condition from both op-
tions fails property (6.1) and therefore is not a valid Cayley-Dickson doubling
product.

The eight Cayley-Dickson doubling products 7
8. The inductive step of the proof of property (6.1)
Suppose (p, q) is an interior point of W2and that (p, q, pq) implies that
(q, pq, p) and (pq, p, q). For which of the sixteen remaining doubling products
does property (6.1) follow for all four successors of (p, q)?
For each of the following four pairs of triple conditions only conditions
from one of each pair may follow from the conditions (p, q, pq), (q, pq, p),
(pq, p, q) and a valid doubling product.
I. Contradictory conditions Aand ˜
A
(2p, 2q, 2pq) and (2q, 2pq , 2p) and (2pq, 2p, 2q) (A)
or
(2q, 2p, 2pq) and (2p, 2pq, 2q) and (2pq, 2q, 2p) ( ˜
A)
II. Contradictory conditions Band ˜
B
(2p, 2q+ 1,2pq + 1) and (2q+ 1,2pq + 1,2p) and (2pq + 1,2p, 2q+ 1) (B)
or
(2q+ 1,2p, 2pq + 1) and (2p, 2pq + 1,2q+ 1) and (2pq + 1,2q+ 1,2p) ( ˜
B)
III. Contradictory conditions Cand ˜
C
(2p+ 1,2q, 2pq + 1) and (2q, 2pq + 1,2p+ 1) and (2pq + 1,2q, 2q) (C)
or
(2q, 2p+ 1,2pq + 1) and (2p+ 1,2pq + 1,2q) and (2pq + 1,2q, 2p+ 1) ( ˜
C)
IV. Contradictory conditions Dand ˜
D
(2p+ 1,2q+ 1,2pq) and (2q+ 1,2pq, 2p+ 1) and (2pq, 2p+ 1,2q+ 1) (D)
or
(2q+ 1,2p+ 1,2pq) and (2p+ 1,2pq, 2q+ 1) and (2pq, 2q+ 1,2p+ 1) ( ˜
D)
Suppose (p, q) is an interior point of W2and that (p, q, pq), (q, pq, p),
(pq, p, q).
1.
e2pe2q= (ep,0)(eq,0)
=((epeq,0) = ( epq,0) = e2pq for f2, f3, f6, f7
(−epeq,0) = (−epq ,0) = −e2pq for f0, f1, f4, f5
(8.1)
So (2p, 2q, 2pq) follows from conditions Afor f2,f3,f6,f7and
(2q, 2p, 2pq) follows from conditions ˜
Afor f0,f1,f4,f5.
2.
e2qe2pq = (eq,0)(epq,0)
=((eqepq ,0) = ( ep,0) = e2pfor f2, f3, f6, f7
(−eqepq,0) = (−ep,0) = −e2pfor f0, f1, f4, f5
(8.2)

8 John W. Bales
So (2q, 2pq, 2p) follows from conditions Afor f2,f3,f6,f7and
(2pq, 2q, 2p) follows from conditions ˜
Afor f0,f1,f4,f5.
3.
e2pq e2p= (epq,0)(ep,0)
=((epqep,0) = ( eq,0) = e2qfor f2, f3, f6, f7
(−epqep,0) = (−eq,0) = −e2qfor f0, f1, f4, f5
(8.3)
So (2pq, 2p, 2q) follows from conditions Afor f2,f3,f6,f7and
(2p, 2pq, 2q) follows from conditions ˜
Afor f0,f1,f4,f5.
4.
e2pe2q+1 = (ep,0)(0, eq)
=((0, epeq) = (0, epq ) = e2pq+1 for g0, g2
(0,−epeq) = (0,−epq ) = −e2pq+1 for g1, g3
(8.4)
So (2p, 2q+ 1,2pq + 1) follows from conditions Bfor g0, g2and
(2q+ 1,2p, 2pq + 1) follows from conditions ˜
Bfor g1, g3.
5.
e2q+1e2pq+1 = (0, eq)(0, epq )
=((eqepq ,0) = ( ep,0) = e2pfor f0, f2, f4, f6
(−eqepq,0) = (−ep,0) = −e2pfor f1, f3, f5, f7
(8.5)
So (2q+ 1,2pq + 1,2p) follows from conditions Bfor f0, f2, f4, f6and
(2pq + 1,2q+ 1,2p) follows from conditions ˜
Bfor f1, f3, f5, f7.
6.
e2pq+1 e2p= (0, epq)(ep,0)
=((0, epq ep) = (0, eq) = e2q+1 for g0, g2
(0,−epqep) = (0,−eq) = −e2q+1 for g1, g3
(8.6)
So (2pq + 1,2p, 2q+ 1) follows from conditions Bfor g0, g2and
(2p, 2pq + 1,2q+ 1) follows from conditions ˜
Bfor g1, g3.
7.
e2p+1e2q= (0, ep)(eq,0)
=((0, epeq) = (0, epq ) = e2pq for g0, g2
(0,−epeq) = (0,−epq ) = −e2pq for g1, g3
(8.7)
So (2pq + 1,2p, 2q+ 1) follows from conditions Cfor g0, g2and
(2p, 2pq + 1,2q+ 1) follows from conditions ˜
Cfor g1, g3.
8.
e2pe2q+1 = (ep,0)(0, eq)
=((0, epeq) = (0, epq ) = e2pq+1 for g0, g2
(0,−epeq) = (0,−epq ) = −e2pq+1 for g1, g3
(8.8)

The eight Cayley-Dickson doubling products 9
So (2p, 2q+ 1,2pq + 1) follows from conditions Cfor g0, g2and
(2q+ 1,2p, 2pq + 1) follows from conditions ˜
Cfor g1, g3.
9.
e2q+1e2pq+1 = (0, eq)(0, epq )
=((eqepq ,0) = ( ep,0) = e2pfor f0, f2, f4, f6
(−eqepq,0) = (−ep,0) = −e2pfor f1, f3, f5, f7
(8.9)
So (2q+ 1,2pq + 1,2p) follows from conditions Cfor f0, f2, f4, f6and
(2pq + 1,2q+ 1,2p) follows from conditions ˜
Cfor f1, f3, f5, f7.
10.
e2p+1e2q+1 = (0, ep)(0, eq)
=((epeq,0) = ( epq ,0) = e2pq for f0, f2, f4, f6
(−epeq,0) = (−epq ,0) = −e2pq for f1, f3, f5, f7
(8.10)
So (2p+ 1,2q+ 1,2pq) follows from conditions Dfor f0, f2, f4, f6and
(2q+ 1,2p+ 1,2pq) follows from conditions ˜
Dfor f1, f3, f5, f7.
11.
e2q+1e2pq = (0, eq)(epq ,0)
=((0, eqepq) = (0, ep) = e2p+1 for g0, g2
(0,−eqepq) = (0,−ep) = −e2p+1 for g1, g3
(8.11)
So (2q+ 1,2pq, 2p+ 1) follows from conditions Dfor g0, g2and
(2pq, 2q+ 1,2p+ 1) follows from conditions ˜
Dfor g1, g3.
12.
e2pq e2p+1 = (pq, 0)(0, ep)
=((0, epq ep) = (0, eq) = e2q+1 for g0, g2
(0,−epqep) = (0,−eq) = −e2q+1 for g1, g3
(8.12)
So (2pq, 2p+ 1,2q+ 1) follows from conditions Dfor g0, g2and
(2p+ 1,2pq, 2q+ 1) follows from conditions ˜
Dfor g1, g3.
These results are compiled in table 8.1 on page 10. Eight more of the
doubling products are eliminated for inconsistency with property 6.1. Thus
the inductive step of the proof of property 6.1 only succeeds for the eight
remaining products.

10 John W. Bales
Table 8.1. Second Elimination of Doubling Products
f0f1f2f3f4f5f6f7
g0A⊠ ⊠
˜
A⊠⊠
B⊠ ⊠ ⊠ ⊠
˜
B⊠ ⊠
C⊠ ⊠ ⊠ ⊠
˜
C⊠ ⊠
D⊠ ⊠ ⊠ ⊠
˜
D⊠ ⊠
g1A⊠ ⊠
˜
A⊠ ⊠
B⊠
˜
B⊠ ⊠ ⊠ ⊠
C⊠ ⊠
˜
C⊠ ⊠ ⊠ ⊠
D⊠ ⊠
˜
D⊠ ⊠ ⊠ ⊠
g2A⊠ ⊠
˜
A⊠ ⊠
B⊠ ⊠ ⊠ ⊠
˜
B⊠ ⊠
C⊠ ⊠ ⊠ ⊠
˜
C⊠ ⊠
D⊠ ⊠ ⊠ ⊠
˜
D⊠ ⊠
g3A⊠ ⊠
˜
A⊠ ⊠
B⊠ ⊠
˜
B⊠ ⊠ ⊠ ⊠
C⊠ ⊠
˜
C⊠ ⊠ ⊠ ⊠
D⊠ ⊠
˜
D⊠ ⊠ ⊠ ⊠
P0: (a, b)(c, d) = (ca −b∗d, da∗+bc)
P1: (a, b)(c, d) = (ca −db∗, a∗d+cb)
P2: (a, b)(c, d) = (ac −b∗d, da∗+bc)
P3: (a, b)(c, d) = (ac −db∗, a∗d+cb)
P⊤
0: (a, b)(c, d) = (ca −bd∗, ad +c∗b)
P⊤
1: (a, b)(c, d) = (ca −d∗b, da +bc∗)
P⊤
2: (a, b)(c, d) = (ac −bd∗, ad +c∗b)
P⊤
3: (a, b)(c, d) = (ac −d∗b, da +bc∗)
The ‘transpose’ symbol is used for the second set of four doubling prod-
ucts since, as will later become apparent, the corresponding product matrices
of unit vectors are transposes of each other.
The product P⊤
3is the one most commonly used, but its transpose P3
is a close second. The author has found no instance of the use of the other
six doubling products.

The eight Cayley-Dickson doubling products 11
All eight of these products result in the complex i=e1= 0,1,0,0,···.
For the first four products, the quaternion j=e2= 0,0,1,0,0,··· and
k=e3= 0,0,0,1,0,···. For their transposes k=e2and j=e3.
9. Recursive definition of structure constants for the
Cayley-Dickson products
For P0, P1, P2and P3,
(1,2n, 2n+ 1) for all n > 0 (9.1)
whereas for P⊤
0, P ⊤
1, P ⊤
2and P⊤
3,
(1,2n+ 1,2n) for all n > 0 (9.2)
The ‘transpositive’ nature of these two properties induces a transpose
relationship between pairs of the eight doubling products.
For all eight of the Cayley-Dickson doubling products, the second quater-
nion property 6.1 holds.
For 0 6=p6=q6= 0
(p, q, r)−→ (q, r, p)−→ (r, p, q) (9.3)
For P0and P⊤
0if 0 6=p6=q6= 0 then
(p, q, r)−→ (2r, 2q, 2p)
−→ (2p, 2q+ 1,2r+ 1)
−→ (2p+ 1,2q, 2r+ 1)
−→ (2p+ 1,2q+ 1,2r) (9.4)
For P1and P⊤
1if 0 6=p6=q6= 0 then
(p, q, r)−→ (2r, 2q, 2p)
−→ (2r+ 1,2q+ 1,2p)
−→ (2r+ 1,2q, 2p+ 1)
−→ (2r, 2q+ 1,2p+ 1) (9.5)
For P2and P⊤
2if 0 6=p6=q6= 0 then
(p, q, r)−→ (2p, 2q, 2r)
−→ (2p, 2q+ 1,2r+ 1)
−→ (2p+ 1,2q, 2r+ 1)
−→ (2p+ 1,2q+ 1,2r) (9.6)

12 John W. Bales
For P3and P⊤
3if 0 6=p6=q6= 0 then
(p, q, r)−→ (2p, 2q, 2r)
−→ (2r+ 1,2q+ 1,2p)
−→ (2r+ 1,2q, 2p+ 1)
−→ (2r, 2q+ 1,2p+ 1) (9.7)
The most commonly used of these eight doubling products is P⊤
3[14]
[5] [10] but P3has also been used [2] [3].
10. Application to octonion basis vectors
Using induction rules 9.1 on the preceding page through 9.7 one can con-
struct all the (p, q, r) for any AN. First using either rule 9.1 or 9.2 on the preceding page
for all nsuch that 2n+ 1 <2Nconstruct all (1, q, r). Then use whichever of
rules 9.4 on the previous page through 9.7 applies to compute the remainder.
For example, using P0, let us compute all the quaternion triplets for the
octonions O=A3. Applying rule 9.2 on the previous page for n= 1,2,3 gives
the quaternion triplets (1,2,3),(1,4,5) and (1,6,7). The remaining four can
be obtained from (1,2,3) and rule 9.4 on the preceding page: (2,6,4),(2,5,7),(6,3,5)
and (4,7,3).
The following are the triplets for P0through P3.
P0: (1,2,3),(1,4,5),(1,6,7),(2,6,4),(7,2,5),(5,6,3),(3,4,7)
P1: (1,2,3),(1,4,5),(1,6,7),(2,6,4),(5,2,7),(7,4,3),(3,6,5)
P2: (1,2,3),(1,4,5),(1,6,7),(2,4,6),(7,2,5),(5,6,3),(3,4,7)
P3: (1,2,3),(1,4,5),(1,6,7),(2,4,6),(5,2,7),(7,4,3),(3,6,5)
Reverse these to obtain the triplets for P⊤
0through P⊤
3.
P⊤
0: (1,3,2),(1,5,4),(1,7,6),(2,4,6),(7,5,2),(5,3,6),(3,7,4)
P⊤
1: (1,3,2),(1,5,4),(1,7,6),(2,4,6),(5,7,2),(7,3,4),(3,5,6)
P⊤
2: (1,3,2),(1,5,4),(1,7,6),(2,6,4),(7,5,2),(5,3,6),(3,7,4)
P⊤
3: (1,3,2),(1,5,4),(1,7,6),(2,6,4),(5,7,2),(7,3,4),(3,5,6)
For P0, P3there are permutations of integers 1 through 7 which take any
triple (p, q, r) through all the triples for that product as well as its transpose.
For example:
P0: (1263457)|(1,2,3),(2,6,4),(6,3,5),(3,4,7),(4,5,1),(5,7,2),(7,1,6)
P3: (1243675)|(1,2,3),(2,4,6),(4,3,7),(3,6,5),(6,7,1),(7,5,2),(5,1,4)

The eight Cayley-Dickson doubling products 13
11. The oriented Fano plane
The products of the basis vectors for the octonions are customarily repre-
sented in the Fano Plane. The shuffle basis produces an especially nice rep-
resentation of the eight doubling products. (See Figure 1 on the next page)
Fano planes are typically numbered in a haphazard fashion although a better
oriented construction exists. To construct an oriented Fano plane draw an
equilateral triangle inscribed with a circle and construct the three altitudes.
Label the center e1. Next label a midpoint of one of the sides e2. Label the
vertex opposite that midpoint e3. Label one of the two remaining midpoints
e4and the vertex opposite that midpoint e5. Label the remaining midpoint
e6and the remaining vertex e7. This results in one of only two versions of
an oriented Fano plane, each being a reflection of the other in one of the
altitudes.
Each of the three sides of the triangle represents a triple (p, q, r).Like-
wise each altitude from a vertex to the midpoint of the opposite side as well
as the circle through the midpoints represents such a triple.
For each of the eight doubling products, the sense of the three sides is
the same–either clockwise (←) around the triangle, or counter-clockwise (→).
If clockwise, then the three sides of the triangle represent (5,2,7),(7,4,3) and
(3,6,5).If counter-clockwise then the three sides represent (7,2,5),(5,6,3)
and (3,4,7).
The circle through the midpoints of the sides represents either (2,4,6)
in the clockwise sense () or (2,6,4) in the counter-clockwise sense ().
The three altitudes may all be in an ‘up’ sense from base to vertex (↑)
or may all be in a ‘down’ sense from vertex to base (↓). So the altitudes must
either be (1,2,3),(1,4,5),and (1,6,7) or (1,3,2),(1,5,4),and (1,7,6). All
altitudes must have the same sense. See Figure 1 on the following page for a
breakdown of all the modes of the eight Cayley-Dickson doubling products.
This common orientation with regards to directions of the three sides as
well as directions of the altitudes, together with the fact that for each (p, q, r),
ris the ‘bit-wise exclusive or’ of pand q, motivates calling this an ‘oriented’
Fano plane. The sides all have the same orientation and the altitudes all have
the same orientation.
Since the sides may have two senses and the circle may have two senses
and the altitudes may have two senses, all 23= 8 versions of the Cayley-
Dickson products may be accommodated in the one diagram.
For the octonions and for any of the eight Cayley-Dickson doubling
variations, knowing the ‘sense’ of (1,2,3),(2,4,6) and (2,5,7) is sufficient
to recover the sense of all octonion triples (p, q, r) for each product using an
oriented Fano plane.

14 John W. Bales
e7e5
e3
e6
e2
e4
e1
Figure 1. Oriented Fano Plane:
P0: (a, b)(c, d) = (ca −b∗d, da∗+bc)↓→
P1: (a, b)(c, d) = (ca −db∗, a∗d+cb)↓←
P2: (a, b)(c, d) = (ac −b∗d, da∗+bc)↓→
P3: (a, b)(c, d) = (ac −db∗, a∗d+cb)↓←
P⊤
0: (a, b)(c, d) = (ca −bd∗, ad +c∗b)↑←
P⊤
1: (a, b)(c, d) = (ca −d∗b, da +bc∗)↑→
P⊤
2: (a, b)(c, d) = (ac −bd∗, ad +c∗b)↑←
P⊤
3: (a, b)(c, d) = (ac −d∗b, da +bc∗)↑→
12. The twists on W corresponding to the eight doubling
products
For each of the eight Cayley-Dickson doubling products of finite sequences,
its multiplication table is divided into distinct 2 ×2 blocks of the form
"e2re2se2re2s+1
e2r+1e2se2r+1 e2s+1#="(er,0)(es,0) (er,0)(0, es)
(0, er)(es,0) (0, er)(0, es)#

The eight Cayley-Dickson doubling products 15
The results of the four products will vary according to which of the eight
Cayley-Dickson doubling products is used. Results are shown in Table 12.1
on page 15.
Table 12.1. Recursive Product of Units
P0e2se2s+1
e2r(eser,0) (0, ese∗
r)
e2r+1 (0, eres) (−e∗
res,0)
P⊤
0e2se2s+1
e2r(eser,0) (0, eres)
e2r+1 (0, e∗
ser) (−ere∗
s,0)
P1
e2r(eser,0) (0, e∗
res)
e2r+1 (0, eser) (−ese∗
r,0)
P⊤
1
e2r(eser,0) (0, eser)
e2r+1 (0, ere∗
s) (−e∗
ser,0)
P2
e2r(eres,0) (0, ese∗
r)
e2r+1 (0, eres) (−e∗
res,0)
P⊤
2
e2r(eres,0) (0, eres)
e2r+1 (0, e∗
ser) (−ere∗
s,0)
P3
e2r(eres,0) (0, e∗
res)
e2r+1 (0, eser) (−ese∗
r,0)
P⊤
3
e2r(eres,0) (0, eser)
e2r+1 (0, ere∗
s) (−e∗
ser,0)
The basis elements e0, e1, e2,· · · are indexed by W={0,1,2,· · ·} which
is a group under the bit-wise ‘exclusive or’ of their binary representations and
for each of the 32 Cayley-Dickson products there is a function ωfrom W×W
to {−1,1}such that for p, q ∈W
epeq=ω(p, q)epq (12.1)
where pq is the group product of pand q. The function ωis called a
‘twist’ on the group Wand turns the set of all finite sequences into a twisted
group algebra.
Since 1 = (1,0) = e0is the identity, it follows that ω(p, 0) = ω(0, q) = 1
for all Cayley-Dickson products, and since for p > 1, epe∗
p=kepk2= 1 and
since epe∗
p=−epep=−ω(p, p)e0= 1 it follows that for p > 1
ω(p, p) = −1 (12.2)
and that for all p
e∗
p=ω(p, p)ep(12.3)
From equation 2.6 on page 2 it follows that
ω(q, p) + ω(p, q) = 0 for 0 6=p6=q6= 0 (12.4)

16 John W. Bales
12.1. The product of e2re2s
Since e2r= (er,0) then e2re2s= (er,0) (es,0) this could be called the case
of b=d= 0.For each of the 32 distinct products, either e2re2s= (eser,0)
or e2re2s= (eres,0). We shall consider the effect each of these alternatives
upon the twist ω.
12.1.1. e2re2s= (eser,0).Since e2re2s=ω(2r, 2s)e2rs and since e2re2s=
ω(s, r)e2rs we may conclude that Whenever e2re2s= (eser,0),
ω(2r, 2s) = ω(s, r) (12.5)
12.1.2. e2re2s= (eres,0).Since e2re2s=ω(2r, 2s)e2rs and since e2re2s=
ω(r, s)e2rs we may conclude that Whenever e2re2s= (eres,0),
ω(2r, 2s) = ω(r, s) (12.6)
12.2. The products e2re2s+1 and e2r+1e2s
Since e2re2s+1 = (er,0) (0, es) this could be called the case of b=c= 0. And
since e2r+1e2s= (0, er) (es,0) that could be called the case of a=d= 0.
An inspection of the 32 alternate products shows that there are only four
distinct possibilities for the products of e2re2s+1 and e2r+1e2s.
12.2.1. e2re2s+1 = (0, eser)and e2r+1e2s= (0, ere∗
s).These two conditions
imply that whenever e2re2s+1 = (0, eser),
ω(2r, 2s+ 1) = ω(s, r) (12.7)
and that whenever e2r+1e2s= (ere∗
s,0),
ω(2r+ 1,2s) = ω(s, s)ω(r, s) = (−ω(r, s) if s > 0
1 otherwise (12.8)
12.2.2. e2re2s+1 = (0, eres)and e2r+1e2s= (0, e∗
ser).These two conditions
imply that whenever e2re2s+1 = (0, eres),
ω(2r, 2s+ 1) = ω(r, s) (12.9)
and that whenever e2r+1e2s= (e∗
ser,0),
ω(2r+ 1,2s) = ω(s, s)ω(s, r) = (−ω(s, r) if s > 0
1 otherwise (12.10)
12.2.3. e2re2s+1 = (0, ese∗
r)and e2r+1e2s= (0, eres).These two conditions
imply that whenever e2re2s+1 = (0, ese∗
r)
ω(2r, 2s+ 1) = ω(r, r)ω(s, r) = (−ω(s, r) if r > 0
1 otherwise (12.11)
and that whenever e2r+1e2s= (eres,0),
ω(2r+ 1,2s) = ω(r, s) (12.12)

The eight Cayley-Dickson doubling products 17
12.2.4. e2re2s+1 = (0, e∗
res)and e2r+1e2s= (0, eser).These two conditions
imply that whenever e2re2s+1 = (0, e∗
res)
ω(2r, 2s+ 1) = ω(s, s)ω(r, s) = (−ω(r, s) if s > 0
1 otherwise (12.13)
and that whenever e2r+1e2s= (eser,0),
ω(2r+ 1,2s) = ω(r, s) (12.14)
12.3. The product e2r+1e2s+1
Since e2r+1 = (0, er) the product
e2r+1e2s+1 could be called the case of a=c= 0.
12.3.1. e2r+1e2s+1 =−(e∗
ser,0).This implies that whenever e2r+1i2s+1 =
−(e∗
ser,0)
ω(2r+ 1,2s+ 1) = −ω(s, s)ω(s, r) = (ω(s, r) if s > 0
−1 otherwise (12.15)
12.3.2. e2r+1e2s+1 =−(ere∗
s,0).This implies that whenever e2r+1e2s+1 =
−(ere∗
s,0))
ω(2r+ 1,2s+ 1) = −ω(s, s)ω(r, s) = (ω(r, s) if s > 0
−1 otherwise (12.16)
12.3.3. e2r+1e2s+1 =−(ese∗
r,0).This implies that whenever e2r+1e2s+1 =
−(ese∗
r,0)
ω(2r+ 1,2s+ 1) = −ω(r, r)ω(s, r) = (ω(s, r) if r > 0
−1 otherwise (12.17)
12.3.4. e2r+1e2s+1 =−(e∗
res,0).This implies that whenever e2r+1e2s+1 =
−(e∗
res,0)
ω(2r+ 1,2s+ 1) = −ω(r, r)ω(r, s) = (ω(r, s) if r > 0
−1 otherwise (12.18)

18 John W. Bales
13. The Eight Variations
The preceding results are summarized in the following tables:
P0: (a, b)(c, d) = (ca −b∗d, da∗+bc)
ω02s2s+ 1
2r ω(s, r)(−ω(s, r) if r > 0
1 otherwise
2r+ 1 ω(r, s)(ω(r, s) if r > 0
−1 otherwise
P⊤
0: (a, b)(c, d) = (ca −bd∗, ad +c∗b)
ω∗
02s2s+ 1
2r ω(s, r)ω(r, s)
2r+ 1 (−ω(s, r) if s > 0
1 otherwise (ω(r, s) if s > 0
−1 otherwise
P1: (a, b)(c, d) = (ca −db∗, a∗d+cb)
ω12s2s+ 1
2r ω(s, r)(−ω(r, s) if r > 0
1 otherwise
2r+ 1 ω(s, r)(ω(s, r) if r > 0
−1 otherwise
P⊤
1: (a, b)(c, d) = (ca −d∗b, da +bc∗)
ω∗
12s2s+ 1
2r ω(s, r)ω(s, r)
2r+ 1 (−ω(r, s) if s > 0
1 otherwise (ω(s, r) if s > 0
−1 otherwise
P2: (a, b)(c, d) = (ac −b∗d, da∗+bc)
ω22s2s+ 1
2r ω(r, s)(−ω(s, r) if r > 0
1 otherwise
2r+ 1 ω(r, s)(ω(r, s) if r > 0
−1 otherwise

The eight Cayley-Dickson doubling products 19
P⊤
2: (a, b)(c, d) = (ac −bd∗, ad +c∗b)
ω∗
22s2s+ 1
2r ω(r, s)ω(r, s)
2r+ 1 (−ω(s, r) if s > 0
1 otherwise (ω(r, s) if s > 0
−1 otherwise
P3: (a, b)(c, d) = (ac −db∗, a∗d+cb)
ω32s2s+ 1
2r ω(r, s)(−ω(r, s) if r > 0
1 otherwise
2r+ 1 ω(s, r)(ω(s, r) if r > 0
−1 otherwise
P⊤
3: (a, b)(c, d) = (ac −d∗b, da +bc∗)
ω∗
32s2s+ 1
2r ω(r, s)ω(s, r)
2r+ 1 (−ω(r, s) if s > 0
1 otherwise (ω(s, r) if s > 0
−1 otherwise
14. The Twist Blocks
The twist tables for each of the eight ωktwist functions can each be subdi-
vided into 2 ×2 blocks with entries +1 or −1. The upper left corner Cof
each twist table for AN,N > 0, corresponds to r=s= 0. For N > 1 the
left side blocks Lcorrespond to r > s = 0. The top side blocks Tcorrespond
to 0 = r < s. The diagonal blocks −Dcorrespond to 0 < r =s. For N > 2,
the interior blocks Ncorrespond to 0 6=r6=s6= 0.Note that Dis shown in
its positive form in Table 14.1 on page 20 but appears as −Din the actual
twist tables. Also notice that Cis the same for all eight doubling products.
Doubling products P0through P3differ from each other only in their values
for Nand products P⊤
0through P⊤
3also differ from each other only in their
values of N. The twist tables for P⊤
kis the transpose of the twist table for
Pk.
Since every pair of basis vectors ep,eqhas a product epeq=ωk(p, q)epq
where pq =p⊕q, the bit-wise ‘exclusive or’ (XOR) of the binary represen-
tations of pand qone only needs a table of values of ωk(p, q) or ω∗
k(p, q) to
recover the multiplication table for any particular 0 ≤k < 4 and any AN.

20 John W. Bales
Table 14.1. Twist Blocks
Twist C L T D N
r=s= 0 r > s = 0 s > r = 0 r=s > 0 0 6=r6=s6= 0
ω0"1 1
1−1# " 1−1
1 1 # " 1 1
1−1# " 1−1
1 1 # " −1 1
1 1 #
ω1"1 1
1−1# " 1−1
1 1 # " 1 1
1−1# " 1−1
1 1 # " −1−1
−1−1#
ω2"1 1
1−1# " 1−1
1 1 # " 1 1
1−1# " 1−1
1 1 # " 1 1
1 1 #
ω3"1 1
1−1# " 1−1
1 1 # " 1 1
1−1# " 1−1
1 1 # " 1−1
−1−1#
ω∗
0"1 1
1−1# " 1 1
1−1# " 1 1
−1 1 # " 1 1
−1 1 # " −1 1
1 1 #
ω∗
1"1 1
1−1# " 1 1
1−1# " 1 1
−1 1 # " 1 1
−1 1 # " −1−1
−1−1#
ω∗
2"1 1
1−1# " 1 1
1−1# " 1 1
−1 1 # " 1 1
−1 1 # " 1 1
1 1 #
ω∗
3"1 1
1−1# " 1 1
1−1# " 1 1
−1 1 # " 1 1
−1 1 # " 1−1
−1−1#
15. Geometry of the ωtwist tables
The ωtables for the eight doubling products are highly periodic when using
the shuffle basis. However, the two most common doubling products ω3and
ω∗
3have a somewhat bewildering pattern to the eye. This is not the case for
ω1and ω2or their transposes. The pattern for ω1‘alternates’ whereas as
N→ ∞, the ω2table for AN+1 has the same appearance as the table for
AN. Note that for any ωtable the table for ANalways forms the upper left
quadrant of the table for AN+1 .
In Figures 2–5 we see visual representations of ω0through ω3for A4
and A7respectively, with dark gray representing ω(p, q) = +1 and light
gray representing ω(p, q) = −1. The upper left square in all images has
coordinates (p, q) = (0,0) and the lower right corner has coordinates (p, q ) =
2N−1,2N−1.

The eight Cayley-Dickson doubling products 21
Figure 2. ω0twist map for A4and A7
Figure 3. ω1twist map for A4and A7

22 John W. Bales
Figure 4. ω2twist map for A4and A7
Figure 5. ω3twist map for A4and A7

The eight Cayley-Dickson doubling products 23
16. The twist tree for ω2
In [3] it was demonstrated how to use a tree diagram to determine ω3(p, q)
for any two non-negative integers pand q. Figure 16.1 is the twist tree for ω2.
To use the tree to find ω2(p, q) for non-negative integers pand qthey
must first be represented in their binary form. To illustrate, let p= 93 =
1011101Band let q= 37 = 0100101B. Next ‘shuffle’ the two bit strings to-
gether to obtain 10,01,10,10,11,00,11. Beginning with the top of the tree
Cwhich represents the top left quadrant of the ω2twist table a binary 1
is an instruction to move to the right branch of the tree and a binary 0 is
an instruction to move to the left branch of the tree. So the first instruc-
tion 10 moves from Cto L. The next instruction 01 moves from Lto −1.
Therefore ω2(93,37) = −1. There is no need to finish traversing the se-
quence 10,01,10,10,11,00,11 further because of the stability of ω2. Thus
e93e37 =−e120 and (37,93,120) is a structure constant for the product P2.
Note that any sequence of instructions which terminates in C, T or L
gives a result of ω2(p, q) = 1 and any which terminates in −Dgives ω2(p, q) =
−1.
This approach to finding the value of the twist function developed in [3]
was found useful by Flaut and Shpakivskyi in their study [7] of holomorphic
functions in generalized Cayley-Dickson algebras.
C
C T L−D
L
L−1L+1
T
T T +1 −1
−D
−D+1 −1−D
Table 16.1. P2Twist Tree (a, b)(c, d) = (ac −b∗d, da∗+bc)
The two dimensional graphs of ω2in Figure 4 suggests that
ω2(p, q) = (1 if 1 <q
p<3
2and
−1 if 2
3<q
p≤1(16.1)
The trees in Table 16.1 suggest that if m > n ≥0 and r, s < 2nthen
ω2(2n+r, 2m+s) = 1 (16.2)
ω2(2m+ 2n+r, 2n+s) = 1 (16.3)
ω2(2m+r, 2m+ 2n+s) = 1 (16.4)

24 John W. Bales
17. Conclusion
We have identified the eight variations of the doubling-products which pro-
duce the Cayley-Dickson algebras and presented an alternate conception of
the order pair of two sequences as consisting of the ‘shuffling’ of the two
sequences. We have demonstrated how the corresponding ‘shuffle basis’ for
Cayley-Dickson algebras reveals the periodic and fractal nature of the twist
underlying the product of the basis vectors of the algebra and identified one
in particular (a, b)(c, d) = (ac −b∗d, da∗+bc) which has a particularly ap-
pealing twist map yet has heretofore not been investigated. We have also
presented another instance of the use of a ‘twist tree’ for determining the
twist function of a Cayley-Dickson algebra. In regard to Octonion algebras
in particular we have suggested an alternate way to label the Fano Plane in
such a way that the structure constants have a consistent orientation in the
plane. It is hoped that the alternate perspectives offered in this paper will
prove useful to other researchers.
Appendix
The author wrote the following bc [11] program for generating the ωtwist
maps in Figures 2 through 5. The code generates a L
A
T
E
X document contain-
ing a PSTricks figure.
For A7it was necessary to increase the LaTeX memory allotment in the
user’s (not the system’s) texmf.cnf file to ‘main_memory = 7500000’.
#TeXTables.bc
define sgn(x,y){
auto p,q,lp,lq;
#Note: lp means p is odd, !lp means p is even
scale=0;
p=x;
q=y;
if(p*q==0) return 1;
lp=p%2;
lq=q%2;
if(p==1 && lq) return -1;
/*The following lines are for omega 0
if(!lp && !lq) return sgn(q/2,p/2);
if(!lp && lq) return -sgn(q/2,p/2);
return sgn(p/2,q/2);
End of omega 0 lines*/
/*The following lines are for omega 1
if(!lp && lq) return -sgn(p/2,q/2);
return sgn(q/2,p/2);
End of omega 1 lines*/
#The following lines are for omega 2
if(!lp && lq) return -sgn(q/2,p/2);
return sgn(p/2,q/2);
#End of omega 2 lines*/

The eight Cayley-Dickson doubling products 25
/*The following lines are for omega 3
if(!lp && !lq) return sgn(p/2,q/2);
if(!lp && lq) return -sgn(p/2,q/2);
return sgn(q/2,p/2);
End of omega 3 lines*/
}
k=7; #Table for Cayley-Dickson Algebra A_k (set desired value)
print "\\documentclass{minimal}\n";
print "\\usepackage{pstricks-ad
if(p==1 && lq) return -1;
/*The following lines are for omega 0
if(!lp && !lq) return sgn(q/2,p/2);
if(!lp && lq) return -sgn(q/2,p/2);
return sgn(p/2,q/2);
End of omega 0 lines*/
/*The following lines are for omega 1
if(!lp && lq) return -sgn(p/2,q/2);
return sgn(q/2,p/2);
End of omega 1 lines*/
#The following lines are for omega 2
if(!lp && lq) return -sgn(q/2,p/2);
return sgn(p/2,q/2);
#End of omega 2 lines*/
print "\\usepackage{graphicx}\n";
print "\\begin{document}\n";
print "\\scalebox{0.8}{\n";
print "\\begin{pspicture}(-13.5,-12.5)(7.5,12.5)\n";
scale=5;
dot=24/2^k; # Sets the size of each square
du=20/2^k; # Sets the distance between squares
v=12+du; # Initializes y coordinate of square
for(p=0;p<2^k;p++){
u=-13-du; # Initializes x coordinate of square
v-=du; # Increments y coordinate of square
for(q=0;q<2^k;q++){
u+=du; # Increments x coordinate of square
if( sgn(p,q)==1 ) print " \\psdot[dotsize=",dot,
"\\psxunit,dotstyle=square,fillcolor=blue]\n
(",u,",",v,")\n" else print " \\psdot[dotsize=",dot,"
\\psxunit,dotstyle=square,fillcolor=yellow]\n(",u,",",v,")
\n"; /* remove last 4 carriage returns before executing */
}
print "\n";
}
print "\\end{pspicture}\n";
print "}\n";
print "\\end{document}";
quit

26 John W. Bales
References
[1] W. Ambrose Structure theorems for a special class of Banach algebras
Trans. Amer. Math. Soc. Vol. 57, (1945) 364-386
[2] J. Baez The Octonions Bul. Am. Math. Soc. Vol. 39, No. 2, (2001)
145-205
[3] J. Bales A tree for computing the Cayley-Dickson twist Missouri J. of
Math. Sci. Vol 21 No. 2 (2009)
[4] Biss, D., Christensen J., Dugger, D., and Isaksen, D. Eigentheory of
Cayley-Dickson algebras In Forum Mathematicum, vol. 21 no. 5, pp.
833-851. 2009.
[5] R. Brown, On generalized Cayley-Dickson algebras Pacific J. Math. Vol.
20 (1967)
[6] R. Busby, H. Smith Representations of twisted group algebras Trans.
Am. Math. Soc. Vol. 149, No. 2, (1970) 503-537
[7] C. Flaut, V. Shpakivskyi, Holomorphic Functions in Generalized
Cayley-Dickson Algebras Springer Basel, Adv. in Appl. Clifford Alge-
bras 25, No. 1, (2015)
[8] Dickson, Leonard E. On quaternions and their generalization and the
history of the eight square theorem Annals of Mathematics (1919): 155-
171
[9] Edwards, C. M., and J. T. Lewis. Twisted group algebras I Communi-
cations in Mathematical Physics 13 no. 2 (1969): 119-130
[10] G. Moreno The zero divisors of the Cayley-Dickson algebras over the
reals Bol. Soc. Mat. Mex. 4, (1998) 13-27
[11] P. Nelson bc: An arbitrary precision calculator language
http://www.gnu.org/software/bc/manual/html_mono/bc.html
(2001)
[12] Nystedt, Patrik, and Johan ¨
Oinert. Simple rings and degree maps Jour-
nal of Algebra 401 (2014): 201-219.
[13] Reynolds, William F. Twisted group algebras over arbitrary fields. Illi-
nois J. Math. 15 (1971), no. 1 91-103
[14] R. Schafer On the algebras formed by the Cayley-Dickson process Amer.
J. Math. Vol. 76 (1954) 435-446
John W. Bales
Department of Mathematics(Retired)
Tuskegee University
Tuskegee, AL 36088
e-mail: john.w.bales@gmail.com
Current address: PO Box 210, Waverly, AL 36879