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 ...