Commutation relation of gamma matrices

While going through the third chapter of Peskin’s QFT book, I am stuck at the following proof:

$$[gamma^mu, S^{rhosigma}] = (mathcal{J}^{rhosigma})^mu_{~nu} gamma^nu,$$

where,

$$S^{munu} = frac{i}{4} [gamma^mu, gamma^nu]qquad;qquadqquad(mathcal{J}^{munu})_{alphabeta} = i (delta^mu_alpha delta^nu_beta – delta^mu_beta delta^nu_alpha).$$

Using these above definitions, I found from R.H.S. of the first equation,

$$(mathcal{J}^{rhosigma})^mu_{~nu} gamma^nu = i (delta^{rhomu} gamma^sigma – delta^{sigmamu} gamma^rho)$$

and from L.H.S. of the first equation,

$$[gamma^mu, S^{rhosigma}] = i (g^{rhomu} gamma^sigma – g^{sigmamu} gamma^rho).$$

Clearly there is a big difference between the two sides. One side has the metric tensors, whereas other side has delta functions. I think this will be fine if in the definition of $(mathcal{J}^{munu})_{alphabeta}$ the delta functions are replaced by the metric tensors. It would be great if anyone could shade some light on this.