Checking Invariance under supersymmetry

Question

I was following Brink. Scherk and Schwartz, the variation of the lagrangian w.r.t supersymmetry transformation can be reduced to.
\(\delta L = -igf_{a b c} \bar{\lambda}^a \gamma_\mu \lambda^b \delta A^{\mu c} = gf_{a b c}( \bar{\lambda}^a \gamma_\mu \lambda^b \bar{\alpha}\gamma^\mu\lambda^c - \bar{\lambda}^a \gamma_\mu \lambda^b \bar{\lambda^c}\gamma^\mu \alpha)\)

It says for D = 4 with Majorana(or weyl) condition, for D=6 with Weyl condition and for D=10 with Majorana-Weyl condition this term vanishes.

Please can someone explain this or point me to further references. Fierz identities are used to derive the results, but I really did not understand how they are obtained.

Cross posted at

Comments

Don't get me wrong I know nothing about SUSY.

But perhaps you need to use somehow the commutator of YM, i.e of $[T^a,T^b]=if_{abc}T^c$, not sure how.

Just thinking out loud.

All the best.