the Killing spinor equations (KSEs) which, here, take the form $\delta_{\epsilon} \psi_{\mu}=
\nabla_{\mu}\epsilon=0$, we can consider first their integrability conditions:

$$[\nabla_{\mu},\nabla_{\nu}] \epsilon=
-{\textstyle\frac{1}{4}} R_{\mu\nu}{}^{ab}\gamma_{ab}
\epsilon =0\, ,\,\,\,
\Rightarrow 
R^{\mu}{}_{a}\gamma^{a} \epsilon=0\, .$$

Thus, at least in the case, the KSIs are contained in the integrability conditions. We will see later how to obtain more information from these identities.

Example: $N=2,d=4$ Supergravity.
--------------------------------

This is the next simplest supergravity theory, if we do not consider adding matter supermultiplets to the $N=1$ theory. Its field content is $\{e^{a}{}_{\mu},A_{\mu}, \psi_{\mu}\}$ (but now $ \psi_{\mu}$ is a Dirac spinor, instead of a Majorana spinor as in the $N=1$ case). The bosonic action (Einstein-Maxwell’s) and the equations of motion (Einstein’s and Maxwell’s) are

$$\left. S \right|_{\psi_{\mu}=0}
=\int d^{4}x \sqrt{|g|}\, 
\left[R
-{\textstyle\frac{1}{4}}F^{2}\right]\, , 
\,\,\, \Rightarrow\,\,\,
\left\{ 
  \begin{array}{rcl}
\mathcal{E}_{a}{}^{\mu}(e) & = & 
-2 \{ G_{a}{}^{\mu}
-\frac{1}{2} T_{a}{}^{\mu} 
\}\, , \\ 
& & \\
\mathcal{E}^{\mu}(A) & = &  
\nabla_{\alpha} F^{\alpha\mu}\, . \\
  \end{array}
\right.$$

The supersymmetry transformations are

$$\delta_{\epsilon} e^{a}{}_{\mu} =
-i\bar{\epsilon}\gamma^{a}\psi_{\mu}
+{\rm c.c.}\, ,
\hspace{.5cm}
\delta_{\epsilon} A_{\mu} =
 -2i\bar{\epsilon}\psi_{\mu} 
+{\rm c.c.}\, .
\hspace{.5cm}
\delta_{\epsilon} \psi_{\mu} =
\nabla_{\mu}\epsilon
-{\textstyle\frac{1}{8}}F^{ab}
\gamma_{ab}\epsilon\equiv 
\tilde{\cal D}_{\mu}\epsilon\, .$$

Using the bosonic fields supersymmetry transformations, we find that the KSIs take the form

$$\bar{\epsilon}\{ \mathcal{E}_{a}{}^{\mu} 