following identities in an orthonormal frame

$$\begin{aligned}
\label{eq:ksi1}
\mathcal{E}^{ab} & = & \eta^{a}{}_{0}  \eta^{b}{}_{0}\mathcal{E}^{00}\, ,\\
& & \nonumber \\
\label{eq:ksi2}
\langle\, \mathcal{V}/X \mid \, \mathcal{E}^{a} \, \rangle  & = & 
{\textstyle\frac{1}{4}}|X|^{-1} \mathcal{E}^{00}\delta^{a}{}_{0}\, ,\\
& & \nonumber \\
\label{eq:ksi3}
\langle\, \mathcal{U}^{*}_{i^{*}}\mid \, \mathcal{E}^{a} \, \rangle  & = & 
{\textstyle\frac{1}{2}} e^{-i\alpha}\mathcal{E}_{i^{*}}\delta^{a}{}_{0}\, .\end{aligned}$$

These equations imply directly

$$\mathcal{E}^{0m}=0\, ,\,\,\,\,\,
\mathcal{E}^{mn}=0\, ,\,\,\,\,\,
\langle\, \mathcal{V}\mid \, \mathcal{E}^{m} \, \rangle =0\, ,\,\,\,\,\,
\langle\, \mathcal{U}^{*}_{i^{*}}\mid \, \mathcal{E}^{m} \, \rangle=0\, .$$

Further, the r.h.s. of Eq. (\[eq:ksi1\]) is real, and this leads to two important identities:

$$\begin{aligned}
\label{eq:ksi2-2}
\langle\, \mathcal{I} \mid \, \mathcal{E}^{0} \, \rangle  & = & 0\, ,\\
& & \nonumber \\
\label{eq:ksi2-1}
\mathcal{E}^{00} & = & \pm 4|\langle\, \mathcal{V} \mid \, \mathcal{E}^{0}\,
\rangle|\, .\end{aligned}$$

Attractor equations {#sec-attractors}
-------------------

It is well-known that, in general, the scalar fields of the black-hole solutions of these theories have certain attractor values that depend solely on the electric and magnetic charges and which are attained at the event horizons irrespectively of the chosen asymptotic values [@Ferrara:1995ih; @Strominger:1996kf].[^8] The attractor values are those which extremize a specific function; furthermore, the absolute value squared of the central charge for the attractor values is essentially the horizon area [@Ferrara:1996dd; @Ferrara:1996um]. Here we 