# Computation with bispinors for a compatible pair of pure spinors in N=1 supersymmetric vacua compactified down to 4-dimension

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: $$$$\Phi_{\pm} = \eta_{1+} \otimes \eta_{2 \pm}^{\dagger} = \frac{1}{8} \sum_{k=0}^6 \frac{1}{k!} \left( \eta_{2 \pm}^{\dagger} \gamma_{m_k...m_1}\eta_{1+} \right)dx^{m_1} \wedge ... \wedge dx^{m_k}.$$$$ I am looking for an explicit expression of this pair, using a parametrization for the two spinors: $$$$\eta_{1+} = cos(\psi) \eta_+ + sin(\psi) \chi_+ \qquad \eta_{2+} = icos(\psi) \eta_+ - isin(\psi) \chi_+,$$$$ where $$\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?