I'm reading about twistors from the book of Huggett and Tod: $\textit{ An introduction to twistor theory}$. I'm trying to understand everything and reproduce every equation that comes here. So, naturally, I came to a dead road to me.
In the book they state that
and I can't verify this. I only reach this:
$ \begin{eqnarray} \nabla^{A'(A}\beta^{B)}=\bar\varphi^{A'C'}\nabla^{(B}_{\hspace{.2cm}C'}\alpha^{A)}+{k\over 4}\nabla^{A'(A}\alpha^{B)}-\varphi^B_{\hspace{.2cm}C}\nabla^{A'(C}\alpha^{A)}-\varphi^A_{\hspace{.2cm}C}\nabla^{A'(C}\alpha^{B)}+{1\over 4}\alpha^{(A}\nabla^{B)A'}k+X^{CC'}\nabla^{A'(A}\nabla_{CC'}\alpha^{B)} \end{eqnarray}$
I really don't know how the last two terms are zero. This book is full of typos (for example, in equation 5.17, the last term it really is $-k\alpha^A/4$) so I was hopping that there was another typo... or maybe something about the properties of flat spaces that I'm not considering.
In advance I'm very grateful for any help.
P. S. $\alpha^A$ is any spinor, $k$ is such that $\mathcal L_{\textbf{X}}g=kg$, and naturally $\textbf X$ is a conformal Killing vector field.
