I expect the following relation to be vanishing. But it seems not that obvious.

$\Gamma_{ab}^{\lambda}t^at^b \Gamma_{\lambda c(d)}t^c=0$

where $t^a$ are even ghosts, "$ab$" are indices for matrix element, and $\lambda$ denote different Gamma matrices. The Einstein summation convention is used above, i.e. we will sum over all indices except $d$.

I checked for both 3D and 4D Clifford algebra. The relation above seems to be right. But not sure whether it is generally true.

Does the following equation also vanishes?

$\Gamma_{\lambda a b}t^a t^b C^{\lambda} C^{\alpha}C^{\beta}=0$

where $C^{\lambda}$ are odd ghosts, i.e. $C^{\alpha}C^{\beta}=-C^{\beta}C^{\alpha}$.

The left hand side of the equation above is supposed to be something in $\wedge^2 V$, where $V=\{ C^{\lambda}|\lambda=1,2,\cdots,D \}$. $D$ is the dimension of the space.