# Is there a terminology that generalizes the “real part” and the “imaginary part”?

By the Cayley-Dickson construction quaternions can be written as $q=z+wj$ and octonions can be written as $x=q_1+q_2l$. Is there any generalization of the real part and the imaginary part such that $$\Re(q)=z,\ \Re(x)=q_1,\ \Im(q)=w,\ \Im(x)=q_2.$$

Though we can not distinguish imaginary units in general, in this construction we did "assign" a special imaginary unit for generalization from one algebra to another. I have heard of the words "simplex" and "perplex". Is there any widely accepted terminology?

• Plus I do know that the real and imaginary part have different meanings in those algebras. What I want to know is the concepts I defined above... – erachang Jul 5 '17 at 0:27

I don't believe there is any standard terminology for this particular setup. However, the maps you want to define are a type of projection. For instance, for the case of quaternions you could denote the projection onto $\mathbb C$ by $P_{\mathbb C}$ and the map onto the orthogonal complement by $P_{\mathbb C^\perp}$.