<PERSON> rotation on <PERSON> identities
I'm having trouble performing a <PERSON> rotation back to <PERSON> spacetime ($\eta_{\mu\nu}=(-1,1,1,\dots)$), following page 19 in the lecture notes here.
I have this expression (equation 53 from the notes linked) \begin{equation} 0=-k_{\mu}G^{\alpha,\mu\nu}E(k)-iF{\mu}^{\ \nu}G^{\alpha,\mu}E(k)+k^{\nu}-k{\mu}\delta^{\alpha\nu} \end{equation} where $k$ is the momentum 4-vector, $G_E$ are (<PERSON> transformed) 2-points functions and $$ is a 1-point function.
By <PERSON> rotation from Euclidean to <PERSON> the Euclidean Green functions become retarded ones $G_R$, furthermore (I quote) "each upper index zero carries a factor of $i$ while each lower index zero carries a factor of $-i$. Thus we have $J^0_E\rightarrow iJ^0_M$, $T^{0i}E\rightarrow iT^{0i}_M$, $T^{00}_E\rightarrow -T^{00}_M$, $k^E_0\rightarrow -ik_0^M=i\omega$, and a similar rule for $F^{\mu}{\nu}$."
With this prescription the above equation becomes \begin{equation} 0=-k_{\mu}G^{\alpha,\mu\nu}R(k)+iF{\mu}^{\ \nu}G^{\alpha,\mu}R(k)+k^{\nu}-k{\mu}\eta^{\alpha\nu} \end{equation}
However I have no idea how to obtain this result from the first equation. I'd say that the free indices ($\alpha$ and $\nu$) don't matter, whether they are temporal or spatial index they are the same for all the terms and always carry the same number of $i$ factors for each term, so they cannot be the reason for the extra minus sign that the $F_{\mu}^{\ \nu}$ term gains going from Euclidean to <PERSON> spacetime.
Let's study the first term of the first expression and use the prescription to <PERSON> rotate (I'm suppressing the free indices) \begin{equation} k_{\mu}^EG^{\mu}_E=k_0^EG^0_E+k_i^EG^i_E=(-ik_0^M)(iG^0_R)+k_i^MG^i_M=k_0^MG^0_R+k_i^MG^i_M \end{equation} but then?
In the same spirit I get for the second term \begin{equation} F_{\mu}^{\ \nu}G^{\mu}_E=F_0^{\ \nu}G^0_E+F_i^{\ \nu}G^i_E=(-iF_0^{\ \nu})(iG^0_R)+F_i^{\ \nu}G^i_R=F_0^{\ \nu}G^0_R+F_i^{\ \nu}G^i_R \end{equation}
I'm quite sure thare's something really simple that I'm forgetting. Can anyone help? 818