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I ask this question basically because I need help, just an hint, for a computation with bispinors. The background is string theory / supergravity, where many data of a supersymmetric vacuum can be encoded in a compatibile pair of pure spinors. Consider a compactification down to 4-dimension, $N=1$, where $\eta_{1+}$, $\eta_{2+}$ are two spinors on the internal manifold $M_6$, $(\eta_{1, 2+})^*=\eta_{1,2-}$. I can construct a compatible pair of pure spinors $\Phi_{\pm} = \eta_{1+} \otimes \eta_{2 \pm}^{\dagger}$. Using Fierz identities and Clifford map I can write: \begin{equation} \Phi_{\pm} = \eta_{1+} \otimes \eta_{2 \pm}^{\dagger} = \frac{1}{8} \sum_{k=0}^6 \frac{1}{k!} \left( \eta_{2 \pm}^{\dagger} \gamma_{m_k...m_1}\eta_{1+} \right)dx^{m_1} \wedge ... \wedge dx^{m_k}. \end{equation} I am looking for an explicit expression of this pair, using a parametrization for the two spinors: \begin{equation} \eta_{1+} = cos(\psi) \eta_+ + sin(\psi) \chi_+ \qquad \eta_{2+} = icos(\psi) \eta_+ - isin(\psi) \chi_+, \end{equation} where $\eta_+$ and $\chi_+= z\eta_-$ form an $SU(2)$ structure $(j, \omega, z)$ since they are orthogonal ($z$ is complex vector which is meant to act by Clifford moltiplication $z = z_m \gamma^m$).

Question: At the end of the day, I want to write $\Phi_{\pm}$ in term of $(j, \omega, z)$, the result is $(3.17)$ in (https://arxiv.org/abs/0708.1032). Consider for example $\Phi_+$: there are only terms with $k = even$. For the first one, with $k=2$, I can easily write it in terms of $(j, \omega, z)$; BUT for a term like $k=4$: $ \eta_{2+}^{\dagger} \gamma_{m n p q}\eta_{1+} $, how can I write it in terms of $(j, \omega, z)$, since $j,\omega$ are bilinears with 2 gamma matrices and $z$ has only one? I don't know how to expand a term with 4 gamma matrices (or more) in ones with 2 or 1. Can I write $ \eta_{2+}^{\dagger} \gamma_{m n p q}\eta_{1+} $ as something like $(\eta_{2+}^{\dagger} \gamma_{m n}\eta_{1+})^2 $ or something like that?

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