Projection operator of symmetric tensors

Question

If I have the identity $\mathbf{\delta}$ in the space of symmetric tensors and a projector $P$ how do I act with them on tensors? Where $$\delta_{\mu\nu,\alpha\beta}=1/2(g_{\mu\alpha}g_{\nu\beta}+ g_{\mu\beta}g_{\nu\alpha})$$ and $$P^{\mu\nu}_{\phantom{\nu\mu} \rho\sigma}=\frac{1}{4}g^{\mu\nu}g_{\rho\sigma}$$ are defined with the metric $g$.