= Jacobi sign test for supercharge components
{c}
{title2=$[R_i,R_j]=-i\epsilon_{ijk}R_k$}
Let $[J_i,q_a]=(R_i)_{ab}q_b$, with numerical coefficient matrices and $[J_i,J_j]=i\epsilon_{ijk}J_k$. The <Jacobi identity> then gives $(R_jR_i-R_iR_j)q=i\epsilon_{ijk}R_kq$, 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.
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