&=&\rho \tau^{-1} \lambda^{ab}(\omega_{b,z}-3\psi_b\mu_{,z}-\psi_b\epsilon^{cd}\psi_c\psi_{d,z}) \\
a^a{}_{t,z}&=&-\rho\tau^{-1}\lambda^{ab}(\omega_{b,\rho}-3\psi_b\mu_{,\rho}-\psi_b\epsilon^{cd}\psi_c\psi_{d,\rho}) .\end{aligned}$$ As shown in Appendix \[sec:A\], we can set $A_{\rho}=A_{z}=0$. Therefore it follows from eq. (\[eq:mu\]) that the $t$-component of the gauge potential $A$ is determined by $$\begin{aligned}
A_{t,\rho}&=&\sqrt{3}\left[a^a{}_t\psi_{a,\rho}-\rho \tau^{-1}(\mu_{,z}+\epsilon^{bc}\psi_b\psi_{c,z})\right],\\
A_{t,z}&=&\sqrt{3}\left[a^a{}_t\psi_{a,z}+\rho\tau^{-1}(\mu_{,\rho}+\epsilon^{bc}\psi_b\psi_{c,\rho})\right].\end{aligned}$$ Thus, once we determine $\Phi^A= (\lambda_{ab},\omega_a,\psi_a,\mu)$, we can specify the solutions of the system given originally by the action, eq. (\[action\]), with our Killing symmetry assumption. It turns out that the above equations of motion, eq. (\[eq:scalar\]), for $\Phi^A$ are derived from the following action $$\begin{aligned}
S&=&\int d\rho dz \rho\left[ G_{AB}(\partial\Phi^A)(\partial\Phi^B)\right] 
\nonumber \\
 &=& \int d\rho dz \rho 
     \biggl[\:  
             \frac{1}{4}{\rm Tr}(\lambda^{-1}\partial\lambda\lambda^{-1}
                                             \partial\lambda )
           + \frac{1}{4}\tau^{-2}\partial\tau^2 
           + \frac{3}{2}\partial\psi^T \lambda^{-1}\partial\psi 
\nonumber \\
& &{} \qquad \qquad 
           - \frac{1}{2}\tau^{-1}v^T\lambda^{-1}v 
           - \frac{3}{2}\tau^{-1}(\partial\mu 
                                   +\epsilon^{ab}\psi_a\partial\psi_b)^2 
     \biggr]  \,,
\label{eq:action} \end{aligned}$$ where $v=\partial \omega-\psi(3\partial \mu+\epsilon^{bc}\psi_b\partial \psi_c)$. This action is invariant under the global $G_{2(2)}$ transformation.

Mazur identity {#sec:mazur}
==============

In the proof of uniqueness theorems for four-dimensional charged rotating black holes, a key role was played by a certain global identity—called the Mazur identity. This is also the case for five-dimensional charged rotating black holes. In this section, we present the Mazur type identity for our non-linear sigma models derived in the previous section. The derivation parallels that for the vacuum Einstein 