# Four-brackets (Hodges, Momentum Twistors)

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.