  Some relationships between the arithmetic and the geometry of Lipschitz and
Hurwitz integers are presented. In particular, it is shown that the (ternary)
vector product of a Lipschitz integer $\alpha$ with two other Lipschitz
integers, both orthogonal to $\alpha$, is a left and also a right multiple of
$\alpha$, and that the vector product of two left multiples of $\alpha$ with
any other Lipschitz integer is still a left multiple of $\alpha$. We also
provide new arithmetical proofs for some old results of Gordon Pall, and raise
a geometric problem on the location of some integral quaternions that is
related to the factorization of some integers.
