Consider the equation from CFT, \begin{equation} \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} = f(x)\eta_{\mu\nu}\tag{Yellow book eq. 4.3} \end{equation} 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 \begin{equation} \partial_\rho\partial_{\mu}\epsilon_{\nu} + \partial_\rho\partial_{\nu}\epsilon_{\mu} = \eta_{\mu\nu} \partial_\rho f \end{equation} 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 \begin{equation} \partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\mu\partial_{\rho}\epsilon_{\nu} = \eta_{\rho\mu} \partial_\nu f \end{equation} I can also write this equation as \begin{equation} \eta_{\rho\mu} \partial_\nu f = \partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\nu\partial_{\mu}\epsilon_{\rho} \end{equation} 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, \begin{equation} \partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\nu\partial_{\mu}\epsilon_{\rho} = \partial_\nu\partial_{\rho}\epsilon_{\mu} + \partial_\mu\partial_{\rho}\epsilon_{\nu} \end{equation}
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?
