# Canonical form of a spinor

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Here in this paper http://arxiv.org/pdf/hep-th/0610128v3.pdf, you can see that the authors (upon the use of Spin(3,1) gauge transformation) turned the original very general $\epsilon =\lambda 1 +{\mu}^{i}e^i+\sigma e^{12}$ to 3 canonical forms which are $\epsilon=e^2$, $\epsilon=1+\alpha e^1$ and finally $\epsilon=1+\beta e^2$ (this last orbit represents the Killing spinor for the IWP metric which has a time-like Killing vector as per other paper) meanwhile the other two orbits correspond to plane-waves with null Killing vector. My question is how do we know that $\epsilon=1+\beta e^2$ represents Killing spinor for the IWP metric which has a time-like Killing vector?

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