# Cayley-Dickson form of a Quaternion

It is known that using the Cayley-Dickson construction a quaternion $q$ can be written in a symplectic form as $q=x+\mathbf{i}y$ with $x,y \in \mathbb{C}$.

I read in a couple of references that $x$ is called the simplex-part whereas $y$ is called the perplex-part of the quaternion. Is this widely accepted and if not what is the proper onomatology?

On the same topic is there a standard name for the two Quaternion parts comprising an Octonion?

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Personally I've never read or heard the terms "simplex part" and "perplex part". –  Johannes Hahn Feb 3 '10 at 17:15
I'm with Johannes, I've never seen a simplex-part or perplex-part mentioned anywhere...what references used those terms? –  Charles Siegel Feb 3 '10 at 17:38
I believe that this might be engineering nomenclature, who also use the term "symplectic" in the same sense as in the question. I have never come across this usage of symplectic (although I can see why it is used) or "simplex" and "perplex" parts in the mathematics or mathematical physics literature. –  José Figueroa-O'Farrill Feb 3 '10 at 18:28
Moreover, I would say that the Cayley-Dickson process would write the quaternion as a pair $(x,y)$ of complex numbers and not in the way that it is written in the question. –  José Figueroa-O'Farrill Feb 3 '10 at 18:30

I believe there is a good reason why mathematicians don't use the terminology "simplex-part" and "perplex-part": they are not canonical! Indeed, algebraically there is no way to distinguish the elements $i$, $j$ and $k$ in the quaternion algebra $\mathbb{H}$ (and there are in fact many more elements playing the same rôle).

On the other hand, there is a canonical standard involution on $\mathbb{H}$, namely $$\sigma \colon x = a + bi + cj + dk \mapsto \overline{x} := a - bi - cj - dk,$$ and therefore the decomposition of $a + bi + cj + dk$ into the two parts $a$ and $bi + cj + dk$ is canonical. The part $bi + cj + dk$ is often called the pure part of the element $x$.

(This terminology is also used for octonions, and also for (generalized) quaternion and octonion algebras over arbitrary fields of characteristic different from $2$.)

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googling "perplex part" quaternion yields 18 results, including this question. –  Gerald Edgar May 5 '11 at 12:52
googling "pure part" quaternion on the other hand yields 579 results. It seems that "purely imaginary part" is used more often, since googling "purely imaginary part" quaternion gives 12400 hits. –  Tom De Medts May 5 '11 at 13:30

[1] T. A. Ell and S. J. Sangwine, “Hypercomplex Fourier Transforms of Color Images,” IEEE Transactions on Image Processing, vol. 16, no. 1, pp. 22-35, Jan. 2007.

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