# fierz identity for chiral spinors

Hello, dear forum users.
Sorry for disturbance.
I wanted to check simple fierz identities using Cadabra.

$$\bar{u}(k_{1}) \gamma^\mu P_L u(p_{1})\bar{v}(p_{2})\gamma_{\mu}P_{L}v(k_{2})$$
where
$$P_L=(1+\gamma^5)/2$$
is a projection operator.

I used the following code. Seemingly the code is missing something as Cadabra gives the result neglecting chirality of \lambda and \chi. What should I add or change to get the correct result, taking chirality into account? The result should be something like:

$$\bar{\nu}_e\gamma^\mu (1+\gamma_5)e\bar{e}\gamma_{\mu}(1+\gamma_5)\nu_e=\bar{e}\gamma^\mu (1+\gamma_5)e\bar{\nu_e}\gamma_{\mu}(1+\gamma_5)\nu_e$$.

{m,n,p#}::Indices;
{m,n,p}::Integer(0..3);
\Gamma{#}::GammaMatrix;
\bar{#}::DiracBar;
\psi::Spinor(dimension=4, type=Weyl);
\theta::Spinor(dimension=4, type=Weyl);
\lambda::Spinor(dimension=4, type=Weyl, chirality=Negative);
\chi::Spinor(dimension=4, type=Weyl, chirality=Negative);
ex:=\bar{\theta} \Gamma_{m} \chi \bar{\psi} \Gamma^{m} \lambda;
fierz(ex, $\theta, \lambda, \psi, \chi$);