Talk:Cayley–Dickson construction

message: I understood the mapping for the complex numbers, (a, b), where a and b are real, corresponds to a complex number a + bi (where a and b are real). 
However it's not clear to me how (a, b), where a and b are complex, corresponds to a quaternions p + qi + rj + sk (where p, q, r, s are real).
I'm also curious about the inverse map.  Is there a canonical or obvious way to represent the 1, i, j, k quaternion bases in terms of pairs of complex numbers?
By trial and error, I found that
1 <-> (1, 0)
i <-> (i, 0)
j <-> (0, 1)
k <-> (0, i) 
seems to satisfy i2  =  j2  =  k2  =  -1
and ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j
Should some discussion about mapping and inverse mapping be added to the article?
Dgrinstein (talk) 04:47, 5 May 2010 (UTC)
replies: text: The most convenient way is to interpret  as , which is equivalent to your assignments for i, j and k. (The assignment of 1 to (1,0) is forced.) The article probably ought to mention this. --Zundark (talk) 08:40, 5 May 2010 (UTC)
thread_title:  Mapping Cayley-Dickson representation of quaternions to 1,i,j,k representation? 

message: It seems to me (after thinking about it and trying it) that if you replace the formula (a, b) (c, d) = (a c - d^* b, d a + b c^*)\, with (a, b) (c, d) = (a c + d^* b, d a + b c^*)\,, and apply it repeatedly to the reals, you get the split composition algebras (split-complex numbers, split-quaternions, and split-octonions). If this is true, does it appear in any reliable source, as it would tie nicely together how the Cayley-Dickson construction easily zeroes in on the best algebras (the ones that are actually new, instead of being reducible or matrix rings), and with a minor change zeroes in on the remaining ones which, though not the best, still have some nice mathematical properties. (When I try this on the split-complexes, I only get the split-quaternions: so it seems that swapping the sign anywhere along the path from R brings you to the split algebras, while you can only get to the division algebras if you never switch the sign.) Double sharp (talk) 15:15, 31 March 2016 (UTC)
replies: text: Ah, I found a source for this. Double sharp (talk) 12:07, 1 April 2016 (UTC)
replies: text: McCrimmon (2004) A Taste of Jordan Algebras looks like a fine reference with History and links to related areas ! — Rgdboer (talk) 20:42, 1 April 2016 (UTC)
replies: text: Indeed it does, and it's even available online for free to boot. Double sharp (talk) 16:08, 2 April 2016 (UTC)

text: The section § 2.9 Forms Permitting Composition on pages 62 and 63 has wrong and misleading assertions. First, Hamilton biquaternions are isomorphic to M(2,C), not to the direct sum of H with itself (which is Clifford’s biquaternions, here split-biquaternions). Next, McCrimmon writes "A subsequent flood of (false!) higher dimensional algebras carried names such as quadrinions, quines, pluquaternions, nonions, tettarions, plutonians. Ireland especially seemed a factory for such counterfeit algebras." Compare this rant with our hypercomplex numbers article. On the other hand, the identification of complex numbers as binarions is spot-on and opens the door to broader use and understanding of the seven real composition algebras. — Rgdboer (talk) 20:40, 8 April 2016 (UTC)
thread_title:  Split Cayley-Dickson construction 

message: The last sentence of the first paragraph seems to suggest that all algebras generated by the Cayley-Dickson construction are composition algebras. However, we learn further down that the sedenions are no longer a composition algebra. Hence, it might be better to change the sentence to something weaker such as "The first four algebras in this sequence are composition algebras frequently used in mathematical physics". I will leave it to someone more knowledgeable to implement this change. 141.5.38.53 (talk) 17:18, 11 July 2017 (UTC)
replies: text: Thank you reader in Erlangen for noting this fact. Modification has been made. As this Project needs sharp eyes and succinct expression, you are encouraged to register an account and get a User page and Watchlist. — Rgdboer (talk) 23:03, 11 July 2017 (UTC)
thread_title:  Introduction paragraph (composition property) 

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replies: text: This article has been removed as off topic. Hyperjeff had taken it down. — Rgdboer (talk) 02:05, 22 January 2018 (UTC)
thread_title:  External links modified 

message: Hi. I noticed the article state:
The complex conjugate (a, b)* of (a, b) is given by
{\displaystyle (a,b)^{*}=(a^{*},-b)=(a,-b)} {\displaystyle (a,b)^{*}=(a^{*},-b)=(a,-b)} since a is a real number and its conjugate is just a.
This seems to be assuming familiarity with the complex numbers, which are being constructed here. Isn't this circular reasoning? I made an attempt at making a proof which used just the properties described here, with no reference to complex numbers, but I wasn't sure if it would be useful, and it would be original research probably... 
Thanks for reading. :)
JonathanHopeThisIsUnique (talk) 22:37, 20 January 2018 (UTC)
replies: text: Good point. Still, the construction proceeds in the general fashion that starts with an arbitrary field having the trivial involution for its "conjugation". See, for example, Complex plane#Quadratic spaces for illustration of the point. The section is establishing the binarion stage of the construction. The viewpoint was taken by Hamilton, and built up since then. — Rgdboer (talk) 01:58, 22 January 2018 (UTC)
thread_title:  Question 

message: Reading the cited article of Albert(1942) I found that you are changing a sign in the definition. The correct definition published in the Albert paper is: (a,b)(c,d) = (ac + \gamma d^* b,da + bc^* ). This means that, if you chose \gamma=i^2 (where i is the imaginary unit), then you get the Cayley-Dikson construction. On the other hand, if you chose \gamma=1, you get a different algebra. This is clear for me in the article. The cited article of Schafer(1954) uses the same definition. Moreover, Albert explicitly states that \gamma must be different from zero (this is a necessary condition for his Lema 1). On the contrary, you allows \gamma to be zero. And this could be fine because, then, the Albert generalization also includes dual numbers with this choice. Even if I do not see all the consequences at this moment.
In conclusion, with the correct sign, one can understand the Albert generalization in the following way: let \gamma_i be equal to one of {-1, 0, 1} for i=1,2,3...n (n any integer). This kind of sequences can be called a signature and can be represented also with the signs {-,0,+} only. Then, complex numbers have signature (-), quaternions have signature (-,-), octonions have signature (-,-,-); sedenions have signature (-,-,-,-), etc. Any Cayley-Dickson algebra has a signature with only negatives. On the other hand, split numbers have signature (+) and there can be also split numbers with split components (for which I do not have a name) with signature (+,+), etc. Moreover, dual numbers have signature (0) and there can be also dual numbers with dual numbers as components, this will have signature (0,0), etc. Finally, there can be also numbers with mixed signatures like (-,+) (which, I think, you call split-quaternions in the article) or numbers with signatures like (+,-,-,-).
As I understand things, Schafer shows that all algebras with signatures - or + (for any length and mixed in any order) are flexible in his 1954 article, and he was aware of this fact.
The inclusion of dual numbers in the Albert generalization could be more controversial.
17:35, 13 December 2018 (UTC)
thread_title: Abraham Adrian Albert Generalization and the notion of signature

message: In the section about modified Cayley-Dickson construction, it is said that the split octonions are isomorphic to Zorn(R). There is however no mention of it anywhere in the current version, nor any link.  — Preceding unsigned comment added by 109.172.129.12 (talk) 15:47, 2 January 2019 (UTC)
replies: text: Follow the link:Split-octonion#Zorn's vector-matrix algebra — Rgdboer (talk) 02:12, 3 January 2019 (UTC)

text: The link has been put in the article for easy reference. — Rgdboer (talk) 01:24, 4 January 2019 (UTC)
thread_title:  What is Zorn(R) 

message: As it stands the article is long-winded like a university lecture and repetitive. The synopsis below is suggested as containing important missing information such as propositions with proofs: 
The Cayley—Dickson construction is due to Leonard Dickson in 1919 showing how the Cayley numbers can be constructed as a two-dimensional algebra over quaternions. In fact, starting with a field F, the construction yields a sequence of F-algebras of dimension 2n. For n = 2 it is an associative algebra called a quaternion algebra, and for n = 3 it is an alternative algebra called an octonion algebra.  These instances n= 1,2 and 3 produce composition algebras as shown below. 
The case n = 1  starts with elements (a.b) in F x F and defines the conjugate (a,b)* to be (a*, –b) where a* = a in case n = 1, and subsequently determined by the formula. The essence of the F-algebra lies in the definition of  the product of two  elements (a,b) and (c,d):
:(a,b) \times (c,d) \ = \ (ac - d^*b, \ da + bc^*) .
Proposition 1: For z = (a,b)\ \text{and}\ w = (c,d), the conjugate of the product is w^*z^* = (zw)^*.
:proof: (c^*,-d)(a^*,-b) = (c^*a^* + b^*(-d),\ bc^*-da)\ =\ (zw)^*.
Proposition 2: If the F-algebra is associative and N(z)=zz^*,\ \text{then}\ N(zw) = N(z)N(w).
:proof: N(zw) = (ac-d^*b, da+bc^*)(c^*a^*-b*d, -da -bc^*)\ =\ (aa^* + bb^*)(cc^* + dd^*) + terms that cancel by the associative property.
— Rgdboer (talk) 17:20, 4 August 2020 (UTC)
thread_title: Synopsis

message: The box on the top right should be extended to show the properties of algebras of this type of 32, 64, 128 etc. dimensions. I think that calculations have been made up to 256 dimensions.  — Preceding unsigned comment added by 2A00:23C4:7C87:4F00:747C:5E75:53A5:B945 (talk) 13:25, 28 August 2020 (UTC)
thread_title:  Possible extension 

message: Change 'octonians' in first paragraph of Synopsis to 'octonions' 210.23.160.153 (talk) 05:01, 22 November 2022 (UTC)
replies: text: D.Lazard (talk) 09:32, 22 November 2022 (UTC)
thread_title:  Semi-protected edit request on 22 November 2022