# Inconsistent indices in the differential equation of infinitesimal conformal transformation

Consider the equation from CFT, $$$$\partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} = f(x)\eta_{\mu\nu}\tag{Yellow book eq. 4.3}$$$$ which constrains the form of $$\epsilon_{\mu}$$ and eventually we get different types of conformal transformations in $$d \geq 3$$. If I apply $$\partial_{\rho}$$ on the above equation I would obtain $$$$\partial_\rho\partial_{\mu}\epsilon_{\nu} + \partial_\rho\partial_{\nu}\epsilon_{\mu} = \eta_{\mu\nu} \partial_\rho f$$$$ Now, the Yellow Book on CFT and Introduction to Conformal Field Theory says to permute indices of the above equation and we will get the following equations \begin{align} \partial_\rho\partial_{\mu}\epsilon_{\nu} + \partial_\rho\partial_{\nu}\epsilon_{\mu} &= \eta_{\mu\nu} \partial_\rho f\\ \partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\mu\partial_{\rho}\epsilon_{\nu} &= \eta_{\rho\mu} \partial_\nu f\\ \partial_\mu\partial_{\nu}\epsilon_{\rho} + \partial_\nu\partial_{\mu}\epsilon_{\rho} &= \eta_{\nu\rho} \partial_\mu f \end{align}\tag{BP p.8} The above set of equations is given in Introduction to Conformal Field Theory on Pg. 8.

Now, if I consider the second equation in the above set of equations I will have $$$$\partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\mu\partial_{\rho}\epsilon_{\nu} = \eta_{\rho\mu} \partial_\nu f$$$$ I can also write this equation as $$$$\eta_{\rho\mu} \partial_\nu f = \partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\nu\partial_{\mu}\epsilon_{\rho}$$$$ where I imported the structured from $$\eta_{\mu\nu} \partial_\rho f = \partial_\rho\partial_{\mu}\epsilon_{\nu} + \partial_\rho\partial_{\nu}\epsilon_{\mu}$$ and did the index change $$\mu \to \rho$$, $$\nu \to \mu$$ and $$\rho \to \nu$$.

Now a major inconsistency appears if we compare the last two equations, $$$$\partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\nu\partial_{\mu}\epsilon_{\rho} = \partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\mu\partial_{\rho}\epsilon_{\nu}$$$$

We clearly see that the first two terms in the above equation cancel on both sides, but the second term remains. This means $$\partial_\nu\partial_{\mu}\epsilon_{\rho} = \partial_\mu\partial_{\rho}\epsilon_{\nu}$$. There is no reason for them to be equal as there is no symmetry in the equation. What am I doing wrong here?