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I'm trying to prove the Shouten identity for twistors:

[itex] \langle pq\rangle\langle rs\rangle+\langle pr\rangle\langle sq\rangle+\langle ps\rangle\langle qr\rangle=0 [/itex]

It's easy to show that the LHS here is cyclically symmetric under [itex] q\to r\to s \to q[/itex], and also completely antisymmetric in q,r,s (just use [itex] \langle qr \rangle=-\langle rq \rangle[/itex] etc)

But why does this imply the LHS must be zero?