\quad
\alpha = \eta_2 \varphi_0 +  \varphi_2\, , \quad
\gamma = \eta_3 \varphi_0 +  \varphi_3\, , \quad
\xi = \eta_4 \varphi_0 +  \varphi_4\,,\end{aligned}$$ and we define the light-cone coordinates as $$\begin{aligned}
z^- = \frac{1}{2} \mu R^2 (t-\varphi_0)\, , \quad
z^+ = \frac{1}{2\mu} (t+\varphi_0)\, .
\label{lcc}\end{aligned}$$ By defining $r_1,...,r_4$ such that $$\begin{aligned}
r_1= R \psi\, , \quad
r_2 = R \theta\, ,\quad
r_3 = R \rho \sin\beta\, ,\quad
r_4= R \rho \cos\beta\, .\end{aligned}$$ we can parametrize the eight $z^i$ coordinates in the following way $$\begin{aligned}
\label{coordinates}
z^1+iz^2 = r_1e^{i\varphi_1}\, , \quad z^3+iz^4 = r_2e^{i\varphi_2}\, , \cr
z^5+iz^6 = r_3e^{i\varphi_3}\, , \quad z^7+iz^8 = r_4e^{i\varphi_4}\, .\end{aligned}$$

Writing the background – in terms of the coordinate $z^\pm$ and $z^i$ and taking the Penrose limit by sending $R\to\infty$ while keeping $z^\pm$ and $z^i$ fixed, we obtain the following metric $$\label{eq:dsnoflat}
    \begin{split}
        ds^2=&-4dz^+dz^- + dz^i dz^i - \mu^2 \sum_{k=1}^{4}
        \left(1-\eta_{k}^{2}\right)\left[\left(z^{2k-1}\right)^2+\left(z^{2k}\right)^2\right]\left(dz^+\right)^2 \\
        &+ 2\mu \sum_{k=1}^{4}\eta_k \left[z^{2k-1}dz^{2k}- z^{2k}dz^{2k-1}\right]dz^+.
    \end{split}$$ and five-form field strength $$\begin{aligned}
\label{eq:F5z}
F_{(5)} = 2 \mu \,dz^+ \left(dz^1 dz^2 dz^3 dz^4 + dz^5 dz^6 dz^7 dz^8 \right)\, .\end{aligned}$$ We see that by setting the parameters $\eta_k$’s all to zero, we precisely recover the pp-wave background derived in  [@Blau:2002mw; @Berenstein:2002jq]. In this sense, the background 