, seen in projection: {{bulleted list As an additive
group,
H is
free abelian with generators {{nowrap|{(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. The Lipschitz quaternions
L form an index 2 sublattice of
H. The
group of units in
L is the
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) / 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 (in the sense of
ring theory) in the
division ring of quaternions with
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 ideals 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. ==The lattice of Hurwitz quaternions==