Talk:Biquaternion

message: There seems to be a conflict of terminology with regards to the name biquaternion. Hamilton appears to have used the term to mean a quaternion with complex coefficients (i.e. C⊗H), while Clifford (in Preliminary Sketch of Biquaternions, 1873) uses the term to mean an algebra isomorphic to H⊕H, which follows the quaternions in the sequence of Clifford algebras:
:R → C → H → H⊕H → ...
The complexified quaternions are not isomorphic to Clifford's biquaternions. This page presently discusses Hamilton's notion, while the German version of the page discuss's Clifford's notion. Some mention should be made of the conflict. I'm not sure which term is more commonly used. -- Fropuff 21:43, 19 February 2006 (UTC)
thread_title: Conflict of terminology

message: This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:44, 10 November 2007 (UTC)
thread_title:  WikiProject class rating

message: The tag asking for more inline citations has been removed. This is a mathematical article and assertions are verifiable. If a reader or editor requires further edification, please use this Talk space to make comments or raise questions.Rgdboer (talk) 23:24, 20 January 2013 (UTC)
thread_title:  More footnotes 

message: Without reading this entire article line by line, I was confused by the claim about the Linear representation that "each of these three arrays has a square equal to the negative of the identity matrix."  Let's compute:   
When I square the first array, I get  times the identity matrix:  
:\begin{pmatrix}h & 0\\0 & -h\end{pmatrix} \begin{pmatrix}h & 0\\0 & -h\end{pmatrix} = \begin{pmatrix}h^2 & 0\\0 & h^2\end{pmatrix}
When I first looked, I didn't see the Definition that h is "the square root of minus one".  
By the way:  Is "the square root of minus one" in this context the same as the imaginary unit?  I assume so -- and if so, it might be nice to link to that Wikipedia on imaginary unit.  
I"m making a change of this nature.  
Thanks, DavidMCEddy (talk) 09:10, 14 December 2015 (UTC)
thread_title:  Confusion in "Linear representation" 

message: At the end of the section https://en.wikipedia.org/wiki/Biquaternion#Associated_terminology, there is a mention of a "complex light cone" method that has superseded the biquaternions. Can someone explain what this is? Cheers. — Preceding unsigned comment added by 82.13.14.165 (talk) 05:42, 27 May 2019 (UTC)
replies: text: The move to general relativity brought in differential geometry and different linear algebra. See Newman–Penrose formalism. — Rgdboer (talk) 22:49, 27 May 2019 (UTC)
thread_title:  What is this "complex light cone" method that has superseded the biquaternions? 

message: The notation of q^* and q^\star to denote the 2 different conjugations is bad; they can't even be distinguished if the font is small.
As a replacement, I would suggest these ones:
q^* \to \bar{q} \quad = \gamma^0 q^\dagger \gamma_0,
q^\star \to q^c \quad = \gamma^0 q \gamma_0,
also let (q^\star)^* \to q^\dagger.
These ones will match operations on biquanternions represented as commutators of gamma matrices (shown to the very right above):
h = i\gamma^5,
\mathbf i = \sigma^{jk},
h\mathbf i= \sigma^{0i};
where \sigma^{\mu\nu} = \frac12\left[\gamma^\mu,\gamma^\nu\right] and 
\epsilon_{ijk}=1. 150.135.165.6 (talk) 06:45, 15 April 2023 (UTC)
replies: text: Agreed that the two conjugations needed clearer distinction. \star has been replaced with \bar in the math markup. \overline can be used with longer expressions. The asterisk has been kept for the biconjugate; note that it is the conjugate expressed with the Cayley–Dickson construction for the general case. Thank you for bringing this lack of clarity to notice. — Rgdboer (talk) 23:19, 23 April 2023 (UTC)
thread_title:  Bad conjugation notation 

message: I was looking into the reference article on the table addition made on February 20 added by an IP (not a WP user). The link https://www.naturalspublishing.com/files/published/e71f3zs34zg62q.pdf has this table, but it simply defers to a reference to another article [12] S. J. Sangwine, T. A. Ell, N. Le Bihan, Adv. Appl. Clifford Algebras 21, 607-636 (2011).
I couldn't find that specific published article online, but I did find a well referenced article by the same authors from the prior year 2010 arXiv:1001.0240v1 [math.RA] 1 Jan 2010 Fundamental representations and algebraic properties of biquaternions. The table in that is much different (no way to get from that to the table in this WP Biquaternion), but looks right. Other references I find to biquaternion tables seem closer to this and I am concerned the one on the WP page might be wrong (or unsubstantiated) without further published source detail. 
::Jgmoxness (talk) 20:05, 8 May 2024 (UTC)
replies: text: Multiplicaton is clear from commutivity of h with the three imaginary units, which multiply as in the quaternion group. The table does not contribute to the article, IMO, and is moved here for comment:
A Biquaternion multiplication table Naturalspublishing is shown below:
e_ie_j	e_ie_j	e_j	e_j 	e_j 	e_j 	e_j 	e_j 	e_j 	e_j 	e_j	e_j	e_j	e_j	e_j	e_j	e_j	e_j
e_ie_j	e_ie_j	e_0	e_1 	e_2 	e_3 	e_4 	e_5 	e_6 	e_7 	   	   	   	   	   	   	   	
e_i   	e_0   	e_0	e_1 	e_2 	e_3 	e_4 	e_5 	e_6 	e_7 	   	   	   	   	   	   	   	
e_i   	e_1   	e_1	-e_0	e_3 	-e_2	e_5 	-e_4	e_7 	-e_6	   	   	   	   	   	   	   	
e_i   	e_2   	e_2	-e_3	-e_0	e_1 	e_6 	-e_7	-e_4	e_5 	   	   	   	   	   	   	   	
e_i   	e_3   	e_3	e_2 	-e_1	-e_0	e_7 	e_6 	-e_5	-e_4	   	   	   	   	   	   	   	
e_i   	e_4   	e_4	e_5 	e_6 	e_7 	-e_0	-e_1	-e_2	-e_3	   	   	   	   	   	   	   	
e_i   	e_5   	e_5	-e_4	e_7 	-e_6	-e_1	e_0 	-e_3	e_2 	   	   	   	   	   	   	   	
e_i   	e_6   	e_6	-e_7	-e_4	e_5 	-e_2	e_3 	e_0 	-e_1	   	   	   	   	   	   	   	
e_i   	e_7   	e_7	e_6 	-e_5	-e_4	-e_3	-e_2	e_1 	e_0 	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
e_i   	      	   	    	    	    	    	    	    	    	   	   	   	   	   	   	   	
Opinions requested. — Rgdboer (talk) 01:57, 9 May 2024 (UTC)
thread_title:  The BiQuaternion multiplication table references