Superparity and Curvature of Twisted Octonionic Manifolds Embedded in Higher Dimensional Spaces
Stevens Institute of Technology
711 Hudson Street
Hoboken NJ 07030
A construction from basic principles yields 480 representations of the octonions, and additionally gives 7*480 = 3360 representations of a second algebra, the twisted octonions, which differs from the octonions in its pattern of associativity and lack of multiplicative norm (Catto and Chesley ), and in its possessing zero divisors. The same construction is extended to 16 dimensions to show the existence of multiple nonisomorphic types of sedenions (including the type derived from octonions by the Cayley-Dickson construction). A preliminary classification of the types of sedenions is sketched, and the twisted octonions are shown to be a necessary subalgebra of every type of sedenion. Zero divisors of the twisted octonions are exhibited, illuminating their inescapable presence in the sedenions (Moreno ) and higher 2n-ions. A generalization of the Cayley-Dickson construction is derived, whose action on the quaternions is shown to yield the twisted octonions, and whose action on octonions and twisted octonions is studied. The twists of the twisted octonions are compared to the usual concept of parity, and a new superparity concept is sketched in connection with curved manifolds and nonorientabilty.
1 Octonions and Twisted Octonions
The octonions are an example of a hypercomplex algebra in which multiplication has an underlying triadic structure reflecting quaternionic multiplication. That is, the 7 imaginary units of the octonions are grouped into triads, such that any pair of units belongs to exactly one triad, and the triads have the same multiplication table as the quaternions (henceforth such a grouping of imaginary units into a set of triads will be called a quaternionic grouping). There are 30 distinct ways in which seven objects may be so arranged, and each such distinct arrangement gives a distinct family of representations of the octonions. A complete multiplication table also needs a handedness assigned to each triad: (ei, ej, ek) with i < j < k is righthanded if eiejek = -1, and lefthanded if eiejek = +1.
1.1 XOR Construction Method
The indices 1, 2, 3 for the quaternion imaginary units e1, e2, e3 satisfy 1 XOR 2 = 3. In general, i XOR j = k implies j XOR k = i implies k XOR i = j, mirroring the cyclic nature of quaternion multiplication; also, XOR is commutative and associative, with identity 0, and i XOR i = 0 for any i. For the consecutive integers 0, 1, ..., N = 2n-1 there is thus a natural quaternionic grouping of the N imaginary elements into triads (ei, ej, ei XOR j). Note however that if N is not of the form 2n-1 there will be some indexes i, j <= N such that i XOR j > N, i.e. this representation only works for dimension 2n, with 2n-1 imaginaries. Having used this technique to construct a set of quaternionic groupings of the N = 2n-1 imaginaries, other such groupings can be obtained by applying permutations of the indices. In the case of N = 7, there are 7! such permutations, but any given quaternionic grouping is mapped to itself by 168 of the permutations, so there are 7!/168 = 30 distinct such groupings. An alternate way of deriving 30: Consider (e1, e2, ex), (e1, ey, ez), (e1, ea, eb) as the 3 triads containing element e1. There are 5 choices for x, and let y = 4 if x = 3, otherwise y = 3. Either way, y is fixed by the choice of x, leaving 3 choices for z. There remains a choice between either (e2, ey, ea) or (e2, ey, eb), after which choice all other triads are determined; thus there are 5*3*2 = 30 possible choices.
1.2 A General Construction Method
Any representation of the octonions must be one of the 30 quaternionic groupings together with a particular choice of handedness for each triad. A 7 bit number (the signmask) can be used to represent this choice - the nth most significant bit is set if the nth triad (in lexical order) of the quaternionic grouping is lefthanded. For a given choice of quaternionic grouping and signmask, the resulting multiplication table's associativity pattern is then analyzed. The table represents octonions iff (eiej)ek = - ei(ejek) whenever (ei, ej, ek) is not a quaternionic triad. Violation of this condition occurs in 7 out of 8 values of the signmask, resulting in a representation of the twisted octonions characterized by the loss of multiplicative norm (Catto and Chesley ) and the appearance of zero divisors.
1.3 480 Octonion Representations
The following chart is a compact tabulation of the 480 octonion representations (mappable to representations of the Fano plane). The first number is a decimal index, 0-34, followed by one of the 35 possible triples of distinct imaginary units in ascending lexical order. In rows 0-29 there follows a list of 7 decimal indices, referring to the triads in a particular quaternionic grouping as defined above. There then follow 16 hexadecimal numbers giving the signmask values which yield true octonions; any other signmask value in the range 0-128 gives the twisted octonions, for which there are thus (128-16)*30 = 3360 representations.
0: 1 2 3 0 9 14 20 23 27 28 08 0f 11 16 22 25 3b 3c 43 44 5a 5d 69 6e 70 77
1: 1 2 4 0 9 14 21 22 26 29 0b 0c 12 15 21 26 38 3f 40 47 59 5e 6a 6d 73 74
2: 1 2 5 0 10 13 19 24 27 28 03 04 1a 1d 29 2e 30 37 48 4f 51 56 62 65 7b 7c
3: 1 2 6 0 10 13 21 22 25 30 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d
4: 1 2 7 0 11 12 19 24 26 29 0a 0d 13 14 20 27 39 3e 41 46 58 5f 6b 6c 72 75
5: 1 3 4 0 11 12 20 23 25 30 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76
6: 1 3 5 1 6 14 17 23 27 31 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d
7: 1 3 6 1 6 14 18 22 26 32 01 06 18 1f 2b 2c 32 35 4a 4d 53 54 60 67 79 7e
8: 1 3 7 1 7 13 16 24 27 31 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76
9: 1 4 5 1 7 13 18 22 25 33 08 0f 11 16 22 25 3b 3c 43 44 5a 5d 69 6e 70 77
10: 1 4 6 1 8 12 16 24 26 32 00 07 19 1e 2a 2d 33 34 4b 4c 52 55 61 66 78 7f
11: 1 4 7 1 8 12 17 23 25 33 03 04 1a 1d 29 2e 30 37 48 4f 51 56 62 65 7b 7c
12: 1 5 6 2 5 14 17 21 29 31 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76
13: 1 5 7 2 5 14 18 20 28 32 0a 0d 13 14 20 27 39 3e 41 46 58 5f 6b 6c 72 75
14: 1 6 7 2 7 11 15 24 29 31 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d
15: 2 3 4 2 7 11 18 20 25 34 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d
16: 2 3 5 2 8 10 15 24 28 32 0b 0c 12 15 21 26 38 3f 40 47 59 5e 6a 6d 73 74
17: 2 3 6 2 8 10 17 21 25 34 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76
18: 2 3 7 3 5 13 16 21 30 31 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d
19: 2 4 5 3 5 13 18 19 28 33 00 07 19 1e 2a 2d 33 34 4b 4c 52 55 61 66 78 7f
20: 2 4 6 3 6 11 15 23 30 31 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76
21: 2 4 7 3 6 11 18 19 26 34 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76
22: 2 5 6 3 8 9 15 23 28 33 01 06 18 1f 2b 2c 32 35 4a 4d 53 54 60 67 79 7e
23: 2 5 7 3 8 9 16 21 26 34 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d
24: 2 6 7 4 5 12 16 20 30 32 08 0f 11 16 22 25 3b 3c 43 44 5a 5d 69 6e 70 77
25: 3 4 5 4 5 12 17 19 29 33 0b 0c 12 15 21 26 38 3f 40 47 59 5e 6a 6d 73 74
26: 3 4 6 4 6 10 15 22 30 32 03 04 1a 1d 29 2e 30 37 48 4f 51 56 62 65 7b 7c
27: 3 4 7 4 6 10 17 19 27 34 02 05 1b 1c 28 2f 31 36 49 4e 50 57 63 64 7a 7d
28: 3 5 6 4 7 9 15 22 29 33 0a 0d 13 14 20 27 39 3e 41 46 58 5f 6b 6c 72 75
29: 3 5 7 4 7 9 16 20 27 34 09 0e 10 17 23 24 3a 3d 42 45 5b 5c 68 6f 71 76
30: 3 6 7
31: 4 5 6
32: 4 5 7
33: 4 6 7
34: 5 6 7
Before illustrating the use of this table via examples, a few properties of the signmask sets should be noted. Each line's set of 16 special signmask values is such that any number n, 0 ≤ n < 128 either equals one of these special values, or differs in one bit from exactly one of the special values.
Symbolically, let S be one of the sets of 16 special signmask values (occurring on, say, line N of the table above), and n any integer 0 ≤ n ≤ 127. Then:
n ∉ S ⇒ ∃! s, m with s ∈ S and 0 ≤ m ≤ 6 such that n = s XOR 2m;
a useful C program shows which triad (if any) is distinguished for a given choice of table row and hexadecimal signmask value.
For the quaternionic grouping of line N, using a signmask value of n = s XOR 2m to construct a multiplication table does not give the true octonions (associativity doesn't satisfy (eiej)ek = ei(ejek) iff (ei, ej, ek) a quaternionic triad), but a different algebra in which triad m is "distinguished", i.e. the true octonions can be obtained by reversing the handedness of this triad, and only this triad. Reversing the handedness of two triads of a representation of the true octonions is equivalent to reversing the sign of the common unit of those two triads (giving another representation of the true octonions), followed by reversing the handedness of the third triad having that unit in common, making that third triad the distinguished triad of a representation of the twisted octonions. Reversing the handedness of an arbitrary number of triads in an octonion representation will reduce to reversal of handedness of a single triad or no triads, plus some number of axis reflections. Thus, there are only two distinct types of algebra generated by quaternionic grouping and handedness assignment, the true octonions and the twisted octonions.
As a first example of using the table, consider the top line, with index 0: the quaternionic grouping is described as 0 9 14 20 23 27 28, i.e. the triads
(e1, e2, e3), (e1, e4, e5), (e1, e6, e7), (e2, e4, e6), (e2, e5, e7), (e3, e4, e7), (e3, e5, e6).
Picking 6e as the signmask (again from line 0:) and relating its bits 1101110 to the triads gives the octonions with quaternionic subalgebras
(e1, e2, e3), (e1, e5, e4), (e1, e7, e6), (e2, e6, e4), (e2, e5, e7), (e3, e7, e4), (e3, e6, e5).
The signmask value 3a, with bits 0111010, does not appear in line 0; if it is used the triads are instead
(e1, e2, e3), (e1, e5, e4), (e1, e6, e7), (e2, e6, e4), (e2, e7, e5), (e3, e7, e4), (e3, e5, e6),
a representation of the twisted octonions, with distinguished triad (e1, e2, e3) as indicated by the fact that 3a is 3b XOR 01 (i.e. the only way to get 3a via a single-bit change of any signmask in line 0 is to change 3b's lowest order bit, which bit corresponds to (e1, e2, e3)). The product
(e1+e4)(e2+e7) = e1e2 + e1e7 + e4e2 + e4e7 = e3 - e6 + e6 - e3 = 0
exemplifies the appearance of zero divisors, arising because the distinguished triad has the "wrong" handedness, so that the first e3 term has sign reversed from what it would be for the true octonions. A more general discussion of at least one class of zero divisors will be given next.
1.4 Zero Divisors in Twisted Octonions
The twisted octonions are differentiated from the true octonions by the associativity pattern of the imaginary units. For true octonions
ei(ejek) = (eiej)ek iff (ei, ej, ek) is a quaternionic triad (or two or all of the indices are the same), which occurs in 175 cases. In the twisted octonions there are 96 further cases in which (ei, ej, ek) is not a quaternionic triad but associates anyway. The example above of zero divisors has the form
(ei+ej)(ek+α el) = 0, where α = ±1 and all indices are distinct and > 0. Expanding gives
eiek + α ejel + α eiel + ejek = 0, implying
eiek = -α ejel and ejek = -α eiel (N.B. typo in paper drops -),
hence (using anticommutativity of any two distinct imaginary units)
-α el = (ejek)ei = ej(ekei).
Thus a pair of zero divisors can be constructed for each of the 96 exceptional vanishing associators; indeed, a second pair of zero divisors immediately follows from:
(ei+ej)(ek+α el) = 0 ⇔ (ei-ej)(ek-α el) = 0
Note that each factor must contain 1 imaginary unit from the distinguished triad in order to produce the term with the "wrong" sign. Also note that a zero divisor multiplied by any real number remains a zero divisor. Geometrically, (some) zero divisors are thereby associated with lines through the origin.
Another rearrangement shows
eiej = ± ekel.
Hence a recipe for constructing zero divisors: pick any 2 imaginaries ei, ek from the distinguished triad, and any ej not in the distinguished triad. Picking el to satisfy eiej = ± ekel then guarantees that for appropriate sign choice,
(ei + ej)(ek ± el) = 0
as desired. Note also that for ei, ej so chosen, there are actually 2 choices for ek, each with a corresponding el. Any zero-divisor line therefore annihilates 2 independent lines, or any linear combination thereof i.e. a zero-divisor plane. Extending the example given earlier(XOR triads with signmask 3a):
(e1+e4)(A(e2+e7) + B(e3-e6)) = 0 and
(e1-e4)(A(e2-e7) + B(e3+e6)) = 0 for any real A, B.
Finally, note that there are 3 choices for an ei from the distinguished triad, times 4 choices for an ej not in the distinguished triad, times 2 sign choices in (ei ± ej) to construct a linear zero divisor, so there are 3*4*2 = 24 such zero divisor lines, each with a corresponding zero divisor plane.
Applying the XOR construction above to the case of 15 imaginary units produces a multiplication table complete but for signs, which are determined by handedness assignment to the triads contained in the table. Examination of the table reveals:
As in the case of the octonions, other representations are derivable from the XOR representation by permuting indices. Out of the 15! possible permutations, 8!/2 map the XOR triads onto themselves, leading to the tentative conjecture that there are 2*15!/8! different ways to form the quaternionic groupings of a sedenion multiplication table - certainly there are at least that many. Steiner triples exist for systems of dimension other than 2n, raising the possibility that quaternionic groupings might exist other than those derived from index permutation acting on the XOR-based multiplication tables. Indeed, recent results (3/1/2007) show that there are at least 5 different types of quaternionic groupings, distinguished by the number of heptads, which can be 0, 1, 3, 7, or 15. The XOR construction, and the Cayley-Dickson constructions lead to the variety with 15 heptads.
- 35 quaternionic triads (hence signmask has 35 bits)
- 15 groupings of 7 triads (henceforth called heptads), either octonions or twisted octonions (depending on signmask)
- any 2 intersecting triads generate a heptad
- each imaginary unit is in 7 triads and 7 heptads
- each triad is in 3 heptads
- each triad intersects 18 others, is disjoint with 16 others
- each triad intersects any heptad in 1 or 3 basis elements
For the XOR-based multiplication tables, the observations itemized above can easily be derived from some further XOR properties. If c does not equal a XOR b, then a, b, c, a XOR b, a XOR c, b XOR c, a XOR b XOR c, and 0 are all distinct, and form a closed set under XOR. This generalizes to any set of N integers, no two of which XOR to give a third of the set - the set of all possible XOR combinations of 2 or more set elements (including 0 = a XOR a for any a in the set) is closed and of order 2N.
XOR closure maps to multiplicative closure in the representations considered here, simplifying the identification of triad membership in heptad subalgebras.
2.1 Preliminary Classification of Sedenion Types
Testing each of the 235 values of signmask in the XOR-based multiplication tables and analyzing the associators (eiej)ek - ei(ejek) shows that there are 9 broad classes of sedenions, classified by the nature of the heptads: of the 15 heptads, anywhere from 0 to 8 are true octonions, with the balance being twisted. Below, counts[N] shows how many signmask values give N true octonionic heptads in the corresponding multiplication table:
counts = 4699455488
counts = 9688596480
counts = 10254827520
counts = 6041190400
counts = 2582200320
counts = 817152000
counts = 248299520
counts = 25804800
counts = 2211840
counts = 0
Adding these up gives 235, establishing the fact that (at least for representations derived via permutation from the XOR-based multiplication tables) all sedenion types must include at least 7 twisted octonion subalgebras.
2.2 Refinements in Classification
When embedded in the sedenions, heptads have more subtle properties than simply whether they are twisted or not, and a more refined classification of the sedenion types must take this into account. Each twisted heptad has a distinguished triad, and that triad occurs in 2 other heptads as well, which might be untwisted, or twisted with a different distinguished triad, or twisted with the same distinguished triad. Any heptad has 7 triads - how many of them are distinguished in some other heptad? A partial analysis based on these questions has so far revealed more than 52 types of sedenions. Several different types will emerge in constructions presented below. Moreover, any quaternionic groupings not based on permutations of XOR indices would (if they exist) add whole new families of sedenion types to the classification scheme. Although classification is still an ongoing process and the inventory is far from complete, enough is known to inform observations on properties like zero divisors, etc. in the following sections.
2.3 Zero Divisors in Sedenions
Since all sedenion multiplication tables contain at least some twisted octonion multiplication tables embedded in them, there are at least the zero divisors already described for the twisted octonions. Moreover, sharing of distinguished triads can induce complicated interactions among the sets of zero divisors induced by different heptads. A general survey awaits completion of the classification scheme sketched above.
3 The Cayley-Dickson Construction
The well-known Cayley-Dickson construction produces a 2n+1-dimensional hypercomplex algebra from a 2n-dimensional one by defining a new multiplication in terms of the existing one:
(a, b)(c, d) = (ac - db*, a*d + cb).
An alternate, equivalent version is defined in Moreno :
(a, b)(c, d) = (ac - d*b, bc* + da).
To see the equivalence of these two, recast in terms of adjoining a new imaginary j (corresponding to (0, 1) of the doubled algebra) to the existing lower-dimensional algebra, so that
(a + jb)(c + jd) = (ac - db*) + j(a*d + cb).
Using ja = a*j, and (ab)* = b*a*, this becomes
(a + b*j)(c + d*j) = (ac - db*) + (a*d)*j + (cb)*j =
(ac - db*) + (d*a + b*c*)j.
Now replace b and d with their conjugates:
(a + bj)(c +dj) = (ac - d*b) + (da + bc*)j,
giving the form used by Moreno.
In the canonical mapping, for 1 ≤ i ≤ 2n-1 = N-1, ei ↔ (ei, 0); the newly adjoined imaginary eN ↔ (0, 1), and the elements (0, ei) for 1 ≤ i ≤ 2n-1 are referred to as eN+i. Because for any m < 2n, m+2n = m XOR 2n, extension via the canonical mapping is just the XOR construction, plus a certain particular assignment of signmask.
3.1 Action on the Quaternions and [Twisted] Octonions
It is a straightforward exercise to show that under the canonical mapping, the first variant above of the Cayley-Dickson construction, acting on the quaternions, gives the representation of the octonons given by line 0, signmask 6e of the table in section 1.3; the Moreno variant gives line 0, signmask 44 (hexadecimal). Moreover, by varying the mapping in all possible ways, so that e.g. -e1 ↔ (0,1), +e4 ↔ (e2,0) etc. all 480 possible octonion multiplication tables can be obtained.
Action on the octonions is more complicated to describe, due to the number of choices of octonions or twisted octonions with which to start, and the complexity of the resulting sedenions. In summary, a type of sedenion is obtained in which there are 7 twisted and 8 true octonion heptads. There are 7 different triads that are distinguished, one from each of the twisted heptads (note that in general it's possible for a single triad to be distinguished in more than one twisted heptad, but such is not the case here). These 7 together form one of the true octonionic heptads, and each of the other 7 true octonionic heptads contains exactly one of these distinguished triads. Surprisingly, this is true even if the Cayley-Dickson construction acts on twisted octonions.
3.2 Derivation and Generalization of Cayley-Dickson
It is noteworthy that neither Cayley-Dickson variant discussed so far can produce the twisted octonions from the quaternions. To find a variant that can, start with the most general form, then eliminate candidates which fail to exhibit certain desired properties e.g. (ab)* = b*a*. Multiplying (a,b)(c,d) must give crossterms of general form ac, bd, ad, and bc, but the factors of a particular crossterm might be reversed in order, and/or complex conjugated (one, or both, or neither). To organize the analysis, assign a yes/no value to the following possibilities:
1. bd vs. db; 2. b conjugated; 3. d conjugated;
4. ad vs. da; 5. a conjugated; 6. d conjugated;
7. bc vs. cb; 8. b conjugated; 9. c conjugated;
It is assumed that the first crossterm must be just ac. Changing the sign of the bd term to + is well known to give split algebras with some "imaginary" units squaring to +1, and so won't be considered here. The 9 remaining binary choices then give 512 possible forms to consider. All but 32 (N.B. original paper said "16" here, ignoring terms like d*a +b*c*, in which b and d are both conjugated) are eliminated by not satisfying (ab)* = b*a*. Of the 32 remaining, 2 are already described above, 28 produce multiplication tables for 7 imaginaries in which some triad does not have quaternionic properties, and 2 new variants emerge, which are related to each other in the same way that the usual Cayley-Dickson form and the Morano variant are related. These variants are:
(a, b)(c, d) = (ac - b*d, da* + bc), and
(a, b)(c, d) = (ac - bd*, ad + c*b).
3.3 Action of the New Cayley-Dickson Versions on the Quaternions
Applying the new Cayley-Dickson multiplication formula
(a, b)(c, d) = (ac - b*d, da* + bc) to the quaternions gives twisted octonions of the XOR representation with signmask 3a; comparing with the octonion representation table in section 1.3 shows that since 3a = 3b XOR 1, the distinguished triad is (e1, e2, e3). For the alternate new version, the signmask is 0x10, again indicating that the distinguished triad is (e1, e2, e3).
3.4 Action on the Octonions and Twisted Octonions
Action of the new variant Cayley-Dickson construction on true octonions yields (again surprisingly) the same type of sedenions as already generated by the usual Cayley-Dickson process. However, when acting on twisted octonions, a different type of sedenion is produced, in which all heptads are twisted, and moreover, with a particular pattern of distribution of the distinguished triads among the heptads: there is one triad that is distinguished in 3 different heptads (i.e. is responsible for twisting those heptads), 3 other triads distinguished in 2 heptads, and 6 triads which are distinguished in a single heptad.
Properties of the systems discussed above, like zero divisors, are of interest in e.g. quark theory (Catto ). Particularly interesting is parity. Axis inversion and triad handedness reversal are trivially equivalent in three dimensions (i.e. the imaginary part of the quaternions). We see however that in spaces of 2N-1 dimensions with a triad-based multiplication, inversion of an axis results in reversed handedness of multiple triads, i.e. all that contain the inverted axis as an element. A chiral inversion of some but not all of these triads (a "superparity" inversion) is thus a finer-grained operation on the space, and raises interesting questions, both mathematical and physical. For example, analysis of nonorientable manifolds (e.g. a 2-dimensional Moebius strip embedded in 3 dimensions) is generally cast in terms of axis inversion; is there a comparable construction based on triad inversion? In physics, is there a variant of the CPT theorem that is based on triad inversion rather than axis reflection? If internal symmetries of particles are somehow an embodiment of higher-dimensional spatial symmetries, might round-trip traversal of a triad-nonorientable manifold produce phenomena like neutrino flavor change? The systems studied above might turn out to be totally unrelated to any physical phenomena, yet it is remarkable how often apparently abstract mathematical structures turn out to have deep physical meaning.
Foremost among those deserving thanks is Sultan Catto, without whose aid and encouragement the 1989 paper would never have been written, and whose kind invitation to participate in this year's 26th International Colloquium on Group Theoretical Methods in Physics stimulated the present paper. More inspiration came from the gala event at Columbia University for T.D. Lee's 80th birthday and the 50th anniversary of the discovery of parity violation. This paper is humbly offered as a small tribute. J. Baez's Web version of the paper cited below was a constant guide and inspires posting of this paper and the various computer programs and other techniques, details of proofs etc. at http://captaincomputersensor.net and
- J. Baez The Octonions, arXiv:math.RA/0105155 v4 23 Apr 2002
- S. Catto and D. Chesley Twisted Octonions and their Symmetry Groups, Nuclear Physics B (Proc. Suppl.) 6 (1989) 428-432
- S. Catto Exceptional Projective Geometries and Internal Symmetries, arXiv:hep-th/0302079 v1 (2003)
- G. Moreno The Zero Divisors of the Cayley-Dickson Algebras over the Real Numbers, arXiv:q-alg/9710013 v1 (1997)