both sides by $X$ and expanding the r.h.s. using $\mathcal{V}/X=\mathcal{R}+i\mathcal{I}$ we get

$$\frac{\mathfrak{D}_{\mu}X}{X} =2 i |X|^{2} \langle\, \mathcal{R} \mid 
\nabla_{\mu} \mathcal{R} \, \rangle 
-2\nabla_{\mu}|X|^{2} \langle\, \mathcal{I} \mid \mathcal{R} \, \rangle 
-2|X|^{2} \langle\, \mathcal{I} \mid \nabla_{\mu} \mathcal{R} \, \rangle\, .$$

Now, from Eq. (\[eq:MRI\])

$$-2 \langle\, \mathcal{I} \mid \nabla_{\mu} \mathcal{R} \, \rangle =  
 \nabla_{\mu}|X|^{-2} -2 
\langle\, \mathcal{R} \mid \nabla_{\mu} \mathcal{I} \, \rangle\, ,$$

and we get

$$\frac{\mathfrak{D}_{\mu}X}{X} =2 i |X|^{2} \langle\, \mathcal{R} \mid 
\nabla_{\mu} \mathcal{R} \, \rangle 
-2|X|^{2} \langle\, \mathcal{R} \mid \nabla_{\mu} \mathcal{I} \, \rangle\, .$$

Finally, using

$$\label{eq:RdR=IdI}
 \langle\, \mathcal{R} \mid 
\nabla_{\mu}\mathcal{R} \, \rangle =
 \langle\, \mathcal{I} \mid 
\nabla_{\mu}\mathcal{I} \, \rangle\, ,$$

which is proved in Appendix \[sec-proofs\], we arrive at[^11]

$$\label{eq:dxZ}
\mathfrak{D}_{\mu}X^{-1} =2  \langle\, \mathcal{V}^{*} \mid 
\nabla_{\mu} \mathcal{I} \, \rangle\, .$$

This equation is valid for all supersymmetric configurations in the timelike class. For those considered in this section we arrive at the equation we were looking for:

$$\mathfrak{D}_{\rho}X^{-1} =2  \mathcal{Z}^{*}\, .$$

The real and imaginary parts of this equation are

$$\begin{aligned}
\label{eq:dgrr}
\frac{d(-g_{rr})}{d\rho} & = & 2  \Re{\rm e}(\mathcal{Z}^{*}/X^{*})
=2 \langle\, \mathcal{R} \mid q \, \rangle\, ,\\
& & \nonumber \\ 
\frac{d\alpha}{d\rho} +\mathcal{Q}_{\rho} & = & |X|^{2}
-2\Im{\rm m}(\mathcal{Z}^{*}/X^{*})
=2 \langle\, \mathcal{I} \mid q \, \rangle
=2 \langle\, 