Let $M$ be a manifold. Suppose that $D$ is a Dirac-type operator on a $\mathbb{Z}_2$-graded Clifford module $E\rightarrow M$, in the sense that $D^2$ is a generalised Laplacian. Define the the action of $T^*M$ on $E$ by the formula $$c(df):=[D,f],$$ where $f$ is a smooth function on $M$. It is claimed (Berline-Getzler-Vergne "Heat Kernels and Dirac Operators" page 113) that:
$$c(d(fh))=[D,fh]=fc(dh)+hc(df),$$ $$c(df)^2=\frac{1}{2}[[D,f],[D,f]]=\frac{1}{2}[[D^2,f],f]=-|df|^2.$$
I would like to know how to get these two identities. It also seems to me that $$[[D,f],[D,f]]$$ should be $0$?
Thanks for your help.