Hurwitz quaternion order

The Hurwitz quaternion order is the subring

\mathcal O_{\mathrm{Hur}}=\mathbb Z\,1+\mathbb Z i+\mathbb Z j+\mathbb Z\frac{1+i+j+k}{2}

of the rational Hamilton quaternion algebra. Equivalently, its four coordinates in $1,i,j,k$ are either all integers or all half-integers in $\mathbb Z+1/2$ . It is an integer order, with inherited quaternion multiplication.

Relationship to Lipschitz quaternions

The Lipschitz order has index two in this lattice. The Hurwitz order is maximal in its rational quaternion algebra. Replacing the displayed last basis vector by $(-1+i+j+k)/2$ gives the same lattice, since the two differ by $1$ .

Units and scalar caution

Its unit group consists of $\pm1,\pm i,\pm j,\pm k$ and the sixteen elements $(\pm1\pm i\pm j\pm k)/2$ , with independent signs. There are 24 units. The order itself is an infinite ring of additive rank four, not this finite group and not a rational vector space.

References

John Voight, Quaternion Algebras, Chapter 11: The Hurwitz order, §11.1, equation (11.1.1), Lemma 11.1.2 and equation (11.1.3); §11.2.