as i was saying

the (so-called) fundamental quaternion units
can be represented as
\pm 1 =\pm\begin{pmatrix} 1&0\\0&1\end{pmatrix}, \pm i =\pm\begin{pmatrix} 0&1\\2&0\end{pmatrix}
\pm j =\pm\begin{pmatrix} 1&1\\1&2\end{pmatrix}, \pm k =\pm\begin{pmatrix} 1&2\\2&2\end{pmatrix},

with the scalars of the matrices considered
as elements of \Bbb{F}_3—i.e., 2=-1 etc.

now let
h =\begin{pmatrix}2&1\\0&2\end{pmatrix},
—the “h” is for hurwitz; let g = h^2
(remember everybody is mod-3; i’d better
get that in somewhere if i’m actually
gonna bring *calculations* into play…)
then \pm (that’s plus-or-minus for you
TeX newbies)
1, i, j, k,
h, hi, hj, hk,
g, gi, gj, gk
is HU—or A-four-hat—or SL_2(\Bbb{F}_3)
or what have you. the binary tetrahedral group.

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