How to prove the equation
$$\int \frac{d^4 k}{(2\pi)^4} \frac{l^\mu l^\nu}{D^3} = \int \frac{d^4 k}{(2\pi)^4} \frac{\frac{1}{4}g^{\mu \nu}{}l^2}{D^3}~?\tag{P&S 6.46}$$
so far, every explanation I've seen in textbooks and on this forum (link) involve the use of Lorentz symmetry or the argument that every spatial direction is equivalent, like equation $$I^{11}=I^{22}=I^{33}=...$$
However, in choosing $l_\mu l_\nu$, didn't we select two special directions $\mu$ and $\nu$ ? In that case, it seems that the symmetry is broken, so why can we still use the symmetry argument ?
