Cayley-Dickson form of a Quaternion - MathOverflow

Cayley-Dickson form of a Quaternion

2010-02-03T16:47:36Z

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?

Answer by Ted Shaneyfelt for Cayley-Dickson form of a Quaternion

2011-04-20T21:34:50Z

[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.

Answer by Tom De Medts for Cayley-Dickson form of a Quaternion

2011-05-05T09:33:48Z

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 $2$.)