= Solution
Use the ordered basis
$$
\left(|3/2\rangle,|1/2\rangle,|-1/2\rangle,|-3/2\rangle\right).
$$
The ladder coefficients are $\sqrt3,2,\sqrt3$, so
$$
S_+=\hbar\begin{pmatrix}
0&\sqrt3&0&0\\
0&0&2&0\\
0&0&0&\sqrt3\\
0&0&0&0
\end{pmatrix},
\qquad
S_-=S_+^\dagger.
$$
Therefore
$$
\boxed{
S_x=\frac\hbar2\begin{pmatrix}
0&\sqrt3&0&0\\
\sqrt3&0&2&0\\
0&2&0&\sqrt3\\
0&0&\sqrt3&0
\end{pmatrix}}
$$
and
$$
\boxed{
S_y=\frac\hbar2\begin{pmatrix}
0&-i\sqrt3&0&0\\
i\sqrt3&0&-2i&0\\
0&2i&0&-i\sqrt3\\
0&0&i\sqrt3&0
\end{pmatrix}.}
$$
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