 Jacobi sign test for supercharge components

Let $[J_i,q_a]=(R_i{)}_{ab}{q}_b$, with numerical coefficient matrices and $[J_i,J_j]=i{ϵ}_{ijk}{J}_k$. The Jacobi identity then gives $(R_j{R}_i-R_i{R}_j)q=i{ϵ}_{ijk}{R}_{k}q$, so component coefficient matrices obey the opposite bracket to state-representation matrices. Thus $R_i=-{\sigma }_i/2$ is consistent for an upper-column spinor, whereas $+{\sigma }_i/2$ in that same component prescription is not. Inverse adjoint actions and dual row conventions must be compared with their index contractions intact.