ra.rings and algebras - Cayley-Dickson form of a Quaternion

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 $2$.)