= Solution
Recover the <angular momentum operators> from $J_1=(J_++J_-)/2$ and $J_2=(J_+-J_-)/(2i)$. The <spin-one angular momentum matrices> in the chosen descending-$J_3$ <orthonormal basis> are
$$
\boxed{J_1=\frac\hbar{\sqrt2}\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix},\qquad
J_2=\frac\hbar{\sqrt2}\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix},\qquad
J_3=\hbar\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}.}
$$
They are <self-adjoint operators>, satisfy $[J_i,J_j]=i\hbar\varepsilon_{ijk}J_k$, and give $J^2=2\hbar^2I$, as required for $j=1$.
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