Trinonions, Quaternions, Quinonions, Sextonions, Septonions, Octonions There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions and septonions?
The term sextonions is used in


*

*Bruce W. Westbury, Sextonions and the magic square http://arxiv.org/abs/math/0411428

*J.M. Landsberg, L. Manivel, The sextonions and $E_{7\frac{1}{2}}$ http://arxiv.org/abs/math/0402157
but this 6-dimensional algebra had been studied earlier in


*

*R.H. Jeurissen, The automorphism groups of octave algebras, Doctoral dissertation, University of Utrecht, 1970.

*E. Kleinfeld, On extensions of quaternions, Indian J. Math. 9 (1968) 443–446.


I encountered the term sextonions at http://en.wikipedia.org/wiki/E7.5 following a link from http://cameroncounts.wordpress.com/2013/09/03/e7-5/. 
They are also mentioned in http://math.ucr.edu/home/baez/week260.html
 A: The main thing to know in this area is Hurwitz theorem and Frobenius theorem which are the results that characterize $\Bbb R$ algebras (actually more) with certain properties.
The former says that composition algebras have dimension 1,2,4 or 8, resulting in the reals, complexes, quaternions and octonions respectively. The latter is a subcase asking about division algebras instead, and it says the only dimensions are 1,2,4, (no octonions.)
Beyond that, if you're not asking for any special properties, there are $\Bbb R$ algebras of every dimension, and people are going to name them whatever they please. I know for sure about the sedenions, which are arrived at by a process analogous to passing up through the chain $R\subseteq C\subseteq H\subseteq O$. This process is known as the Cayley-Dickson construction and can be continued indefinitely producing (nonassociative) algebras of dimension $2^n$. 
In Lemma 3.3 of this paper I see they construct sextonions in what looks like a modified Cayley-Hamilton way, so this might be a generalization that yields the pattern you are looking for. This could indeed be a method for producing such a family of algebras, but the naming convention is entirely arbitrary.
I haven't had the pleasure of meeting the sextonions, but they sound... attractive.
As I remember someone once saying on this site: inventing and naming algebras was a popular sport in the 19th century.
A: 
Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions and septonions?

The known 1-2-4-(6)-8 patterns rely on doubling constructions, and later algebras in the sequence having dimensions divisible by the earlier ones.  It's hard to rule out anything definitively because the definitions can be changed if new patterns appear.  I don't think it's known what the correct definition of algebras is that would support Deligne's conjecture on analytic continuation of the "dimension" or "number of elements" parameters seen in formulas, to non-integer values.
The space of parameters in Vogel's plane that might support interesting Lie algebras is explored here
http://arxiv.org/abs/1209.5709
Mkrtchyan finds $E_{7.5}$ and two larger new examples.  The history is partly explained in his paper, see also the Magic Triangle chapter of the book by Cvitanovic, who was working on very related things for years in his own notation ( http://birdtracks.eu ).

The term sextonions was coined in 2004 in 

I thought the term and the idea came from Westbury, circulating in a preprint by 2003, whose updated form is
http://arxiv.org/abs/math/0411428
It is an older observation of several people that the magic square extends to a magic triangle, and the possibility to add another line to the triangle was also noticed at several times and places, but Westbury seems to be the one who proposed to build this line from a more basic octionion-like object of dimension $6$.  There are people on Mathoverflow who would know the history more exactly, but much can be pieced together from the linked sources and the references in the question.
