BRST quantization: Explicit computation request

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.

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.)