TITLE: Is there a relationship between the cross product and quaternion multiplication?
QUESTION [14 upvotes]: I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus).
Motivation for the question:
I've been introduced to some properties of $\varepsilon_{ijk}$, e.g. antisymmetry (i.e. if you swap two indices, then the $\varepsilon_{...}$ is negated).
Also, cyclic permutations of indices are allowed ($\varepsilon_{ijk}=\varepsilon_{kij}=\varepsilon_{jki}$).
These properties seemed familiar: namely, they satisfy two of the conditions for unit quaternion multiplication. When multiplying unit quaternions, $i, j, k$, we may cyclically permute them ($ijk=kij=jki=-1$), and swapping the order in which we multiply unit quaternions negates the result (e.g. $ij=-ji, jk=-kj, ...$).
Question:

Since $\varepsilon_{ijk}$ is used to represent the cross product of two vectors, is there some inherent relationship between the cross product and quaternion multiplication (and if not/so, why?), or is this just a coincidence?

Any links to some already-established result (if it exists) would also be very helpful!

REPLY [4 votes]: See the product formula at the end of this section about quaternions.  Note that (following the same notation) in particular for $v,w\in \mathbb{R}^3$ $$(0,v)(0,w) = (-v\cdot w, v\times w).$$