BRST quantization: Explicit computation request

Question

Following Green, Schwarz and Witten's book on Superstrings, the BRST charge is given by $$Q_B = c^i K_i-\frac{1}{2}f_{ij}^{~~~k}c^ic^jb_k\tag{3.2.4}$$ with $$[K_i, K_j] = f_{ij}^{~~~k}K_k,\tag{3.2.1}$$ $$\{c^i, b_j\}=\delta^{i}_j,\tag{3.2.2}$$ where the $K_i$ are the generator of the gauge group, $c, b$ are ghosts and anti-ghosts respectively.

Now, the action of the BRST charge on a Yang-Mills gauge field $A^i$ goes like $$Q_B A^i = D c^i$$ with $D$ being the covariant derivative. I wish to know, how to derive the last expression starting from the definition of the BRST charge.

Answer 1

Hint: The Dirac conjecture states that the generators $K_i$ of the gauge group are given by the first-class constraints, i.e. Gauss law $(D\cdot E)_i$. OP's last equation then follows by recalling the CCR for the conjugate pair $A^i$ and $E_j$. (Here deWitt's condensed notation is assumed everywhere.)

References:

1. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994.