$$\label{eq:metricd4}
    ds^{2} = 2|X|^{2}(dt+\omega)^{2} -\frac{1}{2|X|^{2}}dx^{i}dx^{i}\, ,
    \hspace{1cm}
    i,j=1,2,3\, ,$$

    where

    $$\label{eq:MRI}
    \frac{1}{2|X|^{2}}=  \langle\,\mathcal{R}\mid \mathcal{I}\, \rangle\, ,$$

    and $\omega$ is a time-independent 1-form on Euclidean 3-dimensional space satisfying the equation

    $$\label{eq:oidi}
    (d\omega)_{mn} =2\epsilon_{mnp}
    \langle\,\mathcal{I}\mid \partial_{p}\mathcal{I}\, \rangle\, .$$

3.  The symplectic vector of field strengths and their duals $F=(F^{\Lambda},\tilde{F}_{\Lambda})$ is given by

    $$\label{eq:esas}
    F = -{\textstyle\frac{1}{2}} \{d[\mathcal{R} \hat{V}] 
    -{}^{\star}[d\mathcal{I}\wedge \hat{V}] \}\, ,
    \hspace{1cm}
    \hat{V}=2\sqrt{2}|X|^{2}(dt+\omega)\, .$$

The Killing spinors of these solutions have the form

$$\epsilon_{I}=X^{1/2}\epsilon_{I\, 0}\, ,
\hspace{1cm}
\partial_{\mu}\epsilon_{I\, 0}=0\, ,
\hspace{1cm}  
\epsilon_{I\, 0} +i\gamma_{0}
  \epsilon_{IJ}\epsilon^{J}{}_{0}=0\, ,$$

which implies

$$\label{eq:constraint}
\epsilon_{I} +i\gamma_{0}e^{i\alpha} \epsilon_{IJ}\epsilon^{J}=0\, ,  
\hspace{1cm}
e^{i\alpha}= (X/X^{*})^{1/2}\, .$$

Observe that we can write

$$X = \frac{\mathcal{L}^{\Lambda}(Z,Z^{*})}{\mathcal{R}^{\Lambda} 
+i\mathcal{I}^{\Lambda}}\, ,$$

for any $\Lambda$.

Killing Spinor Identities {#sec-KSI}
-------------------------

All supersymmetric configurations satisfy the *Killing Spinor Identities* relating the Einstein equations $\mathcal{E}^{\mu\nu}$, the Maxwell equations $\mathcal{E}_{\Lambda}{}^{\mu}$, the Bianchi identities $\mathcal{B}^{\Lambda\, \mu}$ and the scalar equations of motion $\mathcal{E}^{i}$ [@Kallosh:1993wx; @Bellorin:2005hy; @Meessen:2006tu]

$$\begin{aligned}
\label{eq:ksipsi}
\mathcal{E}_{a}{}^{\mu}\gamma^{a}\epsilon_{I} -4i
\langle\, \mathcal{E}^{\mu} \mid \, \mathcal{V}\, \rangle 
\epsilon_{IJ}\epsilon^{J} & = & 0\, ,\\
& & \nonumber \\
\label{eq:ksilambda} 
\mathcal{E}^{i} \epsilon^{I} +2i
\langle\, \not\!\mathcal{E} \mid \, \mathcal{U}^{*\, i}\, \rangle 
\epsilon^{IJ} \epsilon_{J}& = & 0\, ,\end{aligned}$$

where $\mathcal{E}^{\mu}$ is the symplectic vector $(\mathcal{B}^{\Lambda\,
  \mu}, \mathcal{E}_{\Lambda}{}^{\mu})$.

In the timelike case, they lead to the 