Hurwitz quaternion
a quaternion whose components are either all integers or all half-integers
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is
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{\displaystyle H=\left\{a+bi+cj+dk\in \mathbb {H} \mid a,b,c,d\in \mathbb {Z} \;{\mbox{ or }}\,a,b,c,d\in \mathbb {Z} +{\tfrac {1}{2}}\right\}.}
That is, either a, b, c, d are all integers, or they are all halves of odd integers.
H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by Adolf Hurwitz (1919).
A Lipschitz quaternion (or Lipschitz integer; named after Rudolf Lipschitz) is a quaternion whose components are all integers. The set of all Lipschitz quaternions
L
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{\displaystyle L=\left\{a+bi+cj+dk\in \mathbb {H} \mid a,b,c,d\in \mathbb {Z} \right\}}
forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform Euclidean division on them, obtaining a small remainder.
Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings.
Structure of the ring of Hurwitz quaternions
As an additive group, H is free abelian with generators {(1 + i + j + k) / 2, i, j, k}. It therefore forms a lattice in R4. This lattice is known as the F4 lattice since it is the root lattice of the semisimple Lie algebra F4.
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