Spin-one angular momentum matrices ($J^2=2{\hslash }^{2}I$)

In the ordered angular momentum eigenstate basis $(|1,1⟩,|1,0⟩,|1,-1⟩)$, with positive raising coefficients,To derive them, $[J_3,J_{+}]=\hslash J_{+}$ restricts $J_{+}$ to its two adjacent superdiagonal entries. Write these real entries as $A,B\ge 0$. Since $J_{-}=J_{+}^{†}$, the relation $[J_{+},J_{-}]=2\hslash J_3$ reads $\mathrm{diag}(A^2,B^2-A^2,-B^2)=2{\hslash }^2\mathrm{diag}(1,0,-1)$, forcing $A=B=\sqrt{2}\hslash$. Adding and subtracting the ladder operators gives the displayed matrices. Direct multiplication verifies the angular momentum commutation relations and $J^2=2{\hslash }^{2}I$.