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I use the reference from Andrew Hodges, available at https://arxiv.org/abs/0905.1473. I am having trouble understanding his use of the four-bracket. I refer to equation 6 and equation 9, where he states the equivalence of the following spinor-bracket

$$ [ 4 \vert 5+6 \vert 1 \rangle = \frac{\langle 1 3 4 5 \rangle}{\langle 3 4 \rangle \langle 4 5 \rangle}.\tag{14} $$

I'm sure he's derived this result from using previous definitions of (6) and (11), in terms of contractions of the relevant $W$-twistors. I'm also unclear how he's gone from (10) to (11) in this, i.e how he obtains the relation

$$ [12] = \frac{\langle 0 1 2 3 \rangle}{\langle 0 1 \rangle \langle 1 2 \rangle\langle 2 3 \rangle}.\tag{11} $$

I am familiar with the definition of the four-bracket in terms of a determinant, and have got a nice numerical check, but analytically I am struggling to understand his logic.

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