I am reading Martin Schottenloher's A Mathematical Introduction to Conformal Field Theory. On page 15, edition 2, or page 13 edition 1, he writes that for the case conformal killing factor is $0$, which implies
$$ X_{\mu , \nu} + X_{\nu , \mu} = 0.$$
This he claims implies
$$ X^{\mu}_{,\nu} = 0.$$
I do not understand how he reached this claim.
Then further he writes that this implies
$$ X^{\mu}(q) = c^\mu + w^{\mu}_{\nu} q^{\nu}. $$
Which does not make sense at all, as for the above form of $ X^{\mu}(q)$
$$ X^{\mu}_{,\nu} = w^{\mu}_{\nu}.$$
Take the derivative of $X_{\mu,\nu}+X_{\nu,\mu}=0$ by $\alpha$ to get the relation: $$X_{\mu,\nu\alpha} = -X_{\nu,\mu\alpha}$$ This means that for the second derivatives of a Killing field you can swap a component with a derivative and obtain a minus sign. Since the derivatives commute you can see: $$X_{\mu,\nu\alpha} = - X_{\nu,\mu\alpha} = - X_{\nu,\alpha\mu}= X_{\alpha,\nu\mu}= X_{\alpha,\mu\nu} = -X_{\mu,\alpha\nu}=-X_{\mu,\nu\alpha}$$ and hence $X_{\mu,\nu\alpha}=0$ for any $\mu,\nu,\alpha$.