Grassmann-valued SU(2) adjoint bracket ($(X\times Y{)}_a={\epsilon }_{abc}{X}_b{Y}_c$)

For adjoint fields whose coefficients have Grassmann parity, $X\times Y=-(-1{)}^{|X||Y|}Y\times X$. Thus two odd ghost fields have a symmetric adjoint bracket, and $c\times c$ need not vanish. The graded Jacobi identity and this symmetry are the algebraic mechanisms behind BRST nilpotence.