# Four fermion invariants

In calculations of weak interaction processes in the Fermi-theory,
there are some amplitudes of the form:
$$\bar{a}(\gamma_{\alpha} + \lambda \gamma_{\alpha}\gamma_{5}) b \bar{c}(\gamma^{\alpha} + \gamma^{\alpha}\gamma_5)d$$
where a,b,c,d are Dirac-spinors. Now, if this is a Lorentz-scalar. In that case
it should be a linear combination of a vector*vector and axialvector*axialvector parts,
meaning that axialvector*vector parts should give zero, that is:
$$\bar{a}\gamma_{\alpha}\gamma_{5} b \bar{c}\gamma^{\alpha}d = 0$$
should hold. Can someone show this?

In fact I am a bit confused since $$\gamma_5 \gamma_{\alpha} \gamma^{\alpha} = 4 \gamma_5$$, so if i take for example a=b=c=d, that thing
does not seem to vanish, which does not make sense to me.