= Solution
Write $S_i=(\hbar/2)\sigma_i$, with $\sigma_i$ the <Pauli matrices>. Direct multiplication gives $\sigma_x^2=\sigma_y^2=\sigma_z^2=I$, and hence
$$
\boxed{S_x^2+S_y^2+S_z^2=\frac34\hbar^2I.}
$$
For the products in the other order,
$$
\sigma_x\sigma_y=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=i\sigma_z,\qquad \sigma_y\sigma_x=\begin{pmatrix}-i&0\\0&i\end{pmatrix}=-i\sigma_z.
$$
Their difference gives the <angular momentum commutation relations>
$$
\boxed{[S_x,S_y]=\frac{\hbar^2}{4}(2i\sigma_z)=i\hbar S_z.}
$$
The first identity corresponds to the <spin one-half> value $s(s+1)\hbar^2$ with $s=1/2$.
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