A non-commutative ring from SU(2)

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$SU(2)$, 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?

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Typos corrected:It might help to note that the binary octahedral group is isoclinic to ${\rm GL}(2,3)$, which is clear since they are both double covers of $S_{4}$, and both genuine 2-dimensional representations of these groups yield the same projective (in Schur's sense) representation of $S_{4}$. To be specific, ${\rm SL}(2,3)$ embeds the same way in each of these groups, and if we take an involution $t \in {\rm GL}(2,3) \backslash {\rm SL}(2,3)$, we may replace $t$ by $it$, and we now generate the binary octahedral group.  – Geoff Robinson Dec 30 at 21:30
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