Schouten's identity in Ian Aitchison's supersymmetry book

[LiXinghe]

In 'Supersymmetry in Particle Physics, An Elementary Introduction', the author Ian Aitchison used for several times the following identity:
λa(ζ · ρ) + ζa(ρ · λ) + ρa(λ · ζ) = 0.

I know that this identity is called Schouten's Identity, which is correct when all the variables are common numbers. But, since the λ ζ and ρ are Grassmannian variables, we no longer have λ· λ=0. So how can we use this identity here, since the spinors here are anti-commuting?

 

[haushofer]

Schouten is a general identity saying that in D dimensions antisymmetrization over D+1 indices gives zero because you always encounter twice the same index. So write everything out in spinor components :)