Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish
$$\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 = -\frac{3}{2} \bar{l}_1 l_4 \bar{r}_3 r_2 - \frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2 - \frac{3}{2} \bar{l}_1 (-1) l_4 \bar{r}_3 (+1) r_2 $$$$= -\frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2 = 0$$
Why does this operator vanish? Is that true?
Why is the product of tensor currents expressed in tensor and scalar currents only, while for such combinations of Weyl spinors $l_1$, $r_2$, $r_3$ and $l_4$ in the Fierz identity all of them vanish?