2nd quantization of angular momentum

Given a system with Angular Momentum Operators $J_i,i\in \{1,2,3\}$, we define the ladder operators as $J_{\pm }=J_1\pm i{J}_2$. This definition satisfies

1. $[J^2,J_{\pm }]=0$, where $J^2:={\sum }_{i=1}^3{J}_i^2$
2. $J_{\pm }{J}_{\mp }=J^2-J_3^2\pm \hslash J_3$
3. $[J_{+},J_{-}]=2\hslash J_3$
4. $[J_3,J_{\pm }]=\pm \hslash J_{\pm }$