I'm concerned with a (supposedly) simple identity from Guillemin and Sternberg's book Supersymmetry and Equivariant de Rham Theory. However, it seems false to me, and I would like to have some feedback.
We start with the identities \begin{equation} \begin{cases} d\mu^i= \frac{1}{2}c^b_{ij}(d\theta^i)\theta^j-\frac{1}{2}c^b_{ij}\theta^i(d\theta^j)\\ d\theta^a=\mu^a-\frac{1}{2}c^a_{ij}\theta^i\theta^j \end{cases} \end{equation} we want to use the second one to get an easy expression for $d\mu^i$. The $c^b_{ij}$ are structure constants, thus obey the Jacobi identity, commutativity of the product is given by the rules: \begin{equation} \begin{cases} \theta^i\theta^j=-\theta^j\theta^i\\ \mu^i\theta^j=\theta^j\mu^i \end{cases} \end{equation} (product between the $\mu$'s shouldn't be necessary).
My computation: \begin{equation} \begin{split} d\mu^i=&\frac{1}{2}c^b_{ij}(\mu^i-\frac{1}{2}c^i_{pq}\theta^p\theta^q)\theta^j-\frac{1}{2}c^b_{ij}\theta^i(\mu^j-\frac{1}{2}c^j_{pq}\theta^p\theta^q)\\ =&\frac{1}{2}c^b_{ij}\mu^i\theta^j+\frac{1}{2}c^b_{ji}\mu^j\theta^i\\ &-\frac{1}{4}c^b_{ij}c^i_{pq}\theta^p\theta^q\theta^j-\frac{1}{4}c^b_{ji}c^j_{pq}\theta^i\theta^p\theta^q\\ =&c^b_{ij}\mu^i\theta^j - \frac{1}{4}c^b_{ij}(c^i_{pq}-c^i_{qp})\theta^i\theta^p\theta^q\\ =&c^b_{ij}\mu^i\theta^j - \frac{1}{2}c^b_{ij}c^i_{pq}\theta^i\theta^p\theta^q \end{split} \end{equation}
however, according to the author, the $\frac{1}{2}c^b_{ij}c^i_{pq}\theta^i\theta^p\theta^q$ term is not there, and vanishses thanks to the Jacobi identity - but I don't just don't see how to use it without some more input. Any idea?