Lipschitz quaternion order

The Lipschitz quaternion order is the integer order

L=\mathbb Z\oplus\mathbb Z i\oplus\mathbb Z j\oplus\mathbb Z k

in the rational Hamilton quaternion algebra, where $i^2=j^2=k^2=-1$ and $ij=k=-ji$ . Its multiplication is the inherited quaternion multiplication.

Integral rather than rational

The additive group is free of rank four and its rational span is the ambient algebra. It is not a rational vector space: for example $1/2\notin L$ . The multiplication table has integral structure constants, so the lattice is closed under products.

Units and enlargement

The norm-one elements are exactly $\{\pm1,\pm i,\pm j,\pm k\}$ ; these form its eight-element unit group. The Hurwitz order contains $L$ with index two, so $L$ is not maximal.

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.