About the octonions

Question

Given $a\in \mathbb R^3-\{0\}$, then the map $J_a: q\to q\frac{a}{|a|}$ is a complex structure of the quaternions $\mathbb H$.

Now, what about the octonions $\mathbb O$? Do they similarly obtain a complex structure?

Answer 1

Assuming you mean a linear complex structure, yes. Each $a\in\Bbb R^7\setminus\{0\}$ can be interpreted as an "imaginary" octonion, so as to satisfy $a^2=-|a|^2$, with $|a|$ the length of the original vector. Hence$$q\frac{a}{|a|}\frac{a}{|a|}=q\frac{a^2}{|a|^2}=-q.$$