A Killing vector $K^\mu$ is defined as a vector Lie derivative of metric along which vanishes.
\begin{equation}
\mathcal{L}_K g_{\mu\nu}=0, \quad \Longrightarrow \nabla_\mu K_\nu+\nabla_\nu K_\mu=0.
\end{equation}
I guess there is no need to write derivation of this equation explicitly as you can find it everywhere.

Concerning you question about the antisymmetrisation. Lets start with the expression
$$
\begin{align}\nabla_\nu{K}_\mu\dot{x}^\mu\dot{x}^\nu&=\nabla_\mu{K}_\nu\dot{x}^\nu\dot{x}^\mu\\&\Longrightarrow\,(\nabla_\nu{K}_\mu-\nabla_\mu{K}_\nu)\dot{x}^\mu\dot{x}^\nu=0 \tag{1}\end{align}
$$
In this form the last equation is trivial as we contract an antisymmetric tensor with a symmetric one $\dot{x}^\mu\dot{x}^\nu$. However, you can not derive from here the equation
$$
\nabla_\mu K_\nu-\nabla_\nu K_\mu=0
$$
since this is true only upon contraction with a symmetric tensor.

Hence, the Killing equation $\nabla_\mu K_\nu+\nabla_\nu K_\mu=0$ that was used by Blau actually comes from the definition at the beginning of this post. The symmetrisation in the expression $\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu$ as was already mentioned by John comes from contracting with the symmetric tensor $\dot{x}^\nu\dot{x}^\mu$. In details:
\begin{equation}
\begin{aligned}
\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu&=\frac12(\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu+\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu)\\
&=\frac12(\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu+\nabla_\alpha{K}_\beta\dot{x}^\alpha\dot{x}^\beta)\\
&=\frac12(\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu+\nabla_\alpha{K}_\beta\dot{x}^\beta\dot{x}^\alpha)\\
&=\frac12(\nabla_\nu{K}_\mu\dot{x}^\nu\dot{x}^\mu+\nabla_\mu{K}_\nu\dot{x}^\nu\dot{x}^\mu)\\
&=\frac12(\nabla_\nu{K}_\mu+\nabla_\mu{K}_\nu)\dot{x}^\nu\dot{x}^\mu).
\end{aligned}
\end{equation}
Here in the second line I just renamed the indices, in the third the $\dot{x}^\alpha$ and $\dot{x}^\beta$ were permuted and then I renamed the indices again.

This post imported from StackExchange Physics at 2014-05-04 11:40 (UCT), posted by SE-user Edvard