Confusing "Fourier coefficients" to represent 2d spinors in the helicity spinor formalism

Question

Using the following notation (in the spin helicity formalism) when represnting energy-momentum 4-vectors as bispinors

$$\lambda^{\alpha} = p\rangle, \quad \lambda_{\alpha} = \langle p, \quad \tilde{\lambda}_{\dot{\alpha}} = p], \quad \tilde{\lambda}^{\dot{\alpha}} = [p$$

such that

$$p^{\alpha\dot{\alpha}} = p\rangle[p, \quad p_{\alpha\dot{\alpha}} = p]\langle p$$

a two-dimensional spinor $1\rangle$ can be represented as a sum of two other spinors two-dimensional  $2\rangle$ and $3\rangle$ as

$$1\rangle = \frac{\langle1 3 \rangle}{\langle23\rangle} 2\rangle - \frac{\langle1 2 \rangle}{\langle23\rangle} 3\rangle$$

I dont understand the "Fourier coefficients" in this this case, as I would rather have expected them to be $\frac{\langle 12 \rangle}{\langle 11 \rangle}$ to project on the base spinor $2\rangle$ and $\frac{\langle 13 \rangle}{\langle 11 \rangle}$ to project on the base spinor $3 \rangle$.

Can anybody explain what is going on here? Generally, to me this notation is very confusing and not well enough explained in the source I am reading this from ...

Comments

Relevant links: http://isites.harvard.edu/fs/docs/icb.topic1146666.files/IV-3-SpinorHelicity.pdf, http://arxiv.org/abs/1310.5353v1, http://www.niu.edu/spmartin/spinors/DHMspinors_v5.pdf

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The notation is strange (and the formula you cite seems simply wrong).

If I find a paper confusing to read I usually try to read the same material from another more readable source. (Often I read several related papers or books in parallel.) One can usually find the same stuff treated in multiple places, except when the material is very new or very specialized.