Are $i^2 = j^2 = k^2 = ijk = -1$ axioms?

Question

Regarding the quaternions definition:

$$i^2 = j^2 = k^2 = ijk = -1$$

I wounder if these are axioms, i.e. a self-evident truth that requires no proof? If not, how $ijk = -1$ is derived?

Answer 1

These are definitions.

$$i^2=j^2=k^2=-1$$

Also, from definition, we have $ij=k$, hence

$$ijk=k^2=-1.$$