Editing Hurwitz quaternion (section)

==Structure of the ring of Hurwitz quaternions==
[[File:Binary tetrahedral group elements.png|thumb|24 quaternion elements of the [[binary tetrahedral group]], seen in projection:
{{bulleted list
| 1 {{nowrap|order-1}}: 1
| 1 {{nowrap|order-2}}: −1
| 6 {{nowrap|order-4}}: ±''i'', ±''j'', ±''k''
| 8 {{nowrap|order-6}}: (+1±''i''±''j''±''k'')/2
| 8 {{nowrap|order-3}}: (−1±''i''±''j''±''k'')/2}}]]
As an additive [[group (mathematics)|group]], ''H'' is [[free abelian group|free abelian]] with generators {{nowrap|{(1 + ''i'' + ''j'' + ''k'')&hairsp;/&hairsp;2, ''i'', ''j'', ''k''}.}} It therefore forms a [[lattice (group)|lattice]] in '''R'''<sup>4</sup>. This lattice is known as the [[F4 lattice|''F''<sub>4</sub> lattice]] since it is the [[root lattice]] of the [[semisimple Lie algebra]] [[F4 (mathematics)|''F''<sub>4</sub>]]. The Lipschitz quaternions ''L'' form an index 2 sublattice of ''H''.

The [[group of units]] in ''L'' is the [[order of a group|order]] 8 [[quaternion group]] {{nowrap|1=''Q'' = {±1, ±''i'', ±''j'', ±''k''}.}} The group of units in ''H'' is a [[nonabelian group]] of order 24 known as the [[binary tetrahedral group]]. The elements of this group include the 8 elements of ''Q'' along with the 16 quaternions {{nowrap|{(±1 ± ''i'' ± ''j'' ± ''k'')&hairsp;/&hairsp;2},}} where signs may be taken in any combination. The quaternion group is a [[normal subgroup]] of the binary tetrahedral group U(''H''). The elements of U(''H''), which all have norm 1, form the vertices of the [[24-cell]] inscribed in the [[3-sphere]].

The Hurwitz quaternions form an [[order (ring theory)|order]] (in the sense of [[ring theory]]) in the [[division ring]] of quaternions with [[rational number|rational]] components. It is in fact a [[maximal order]]; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an ''integral quaternion'', also form an order. However, this latter order is not a maximal one, and therefore (as it turns out) less suitable for developing a theory of [[left ideal]]s comparable to that of [[algebraic number theory]]. What [[Adolf Hurwitz]] realised, therefore, was that this definition of Hurwitz integral quaternion is the better one to operate with. For a [[non-commutative ring]] such as '''H''', maximal orders need not be unique, so one needs to fix a maximal order, in carrying over the concept of an [[algebraic integer]].