$SU(2)$, which will be regarded here as the group of unit quaternions under multiplication, has 3 conjugacy classes of finite subgroups which don't have cyclic subgroups of index 1 or 2. They are:

of order $24$, the binary tetrahedral group $ T^{ * } \cong SL(2,3)$

of order $120$, the binary icosahedral group $ I^{ * } \cong SL(2,5)$

of order $48$, the binary octahedral group $ O^{ * }$, which doubly covers $S_{4}$ but is not isomorphic to $SL(2, \mathbb{Z}/(4))$ or $GL(2,3)$

The ring of all sums (using quaternion addition) of elements of $ T^{ * }$ is the Hurwitz ring of integral quaternions. The ring of all sums of elements of $ I^{ * } $ is the icosian ring, which was studied by Hamilton and can be identified with the $E_{8}$ root lattice.

My questions are about the ring of sums of elements of $O^{ * }$. I did not see references to it in SPLAG (which even comes close when discussing lattices over the subring $\mathbb{Z}[e^{\frac{2\pi i}{8}}]$, and discusses both the $E_{8}$ and Leech lattices as icosian lattices), and it's hard to look up a structure whose name I don't know. Does anyone know of a reference that explores this ring in non-trivial depth? Does this ring have a standard name?