Shouldn't Minkowski space even have 10 killing vectors (or rather Killing fields, to avoid the sloppy use of math language by physicists that again seems to lead to some confusion ;-)) generating the full orthochronous proper Poincare group (space-time translations, Lorentz boosts, and rotations)?
Let's see. For Minkowski space we can choose coordinates (the usual ##x^{\mu}=(t,\vec{x})## ones) such that
$$g_{\mu \nu}=\eta_{\mu \nu}=\text{const}.$$
The Killing equation thus reads
$$\nabla_{\mu} \xi_{\nu} + \nabla_{\nu} \xi_{\mu}=\partial_{\mu} \xi_{\nu} + \partial_{\nu} \xi^{\mu}=0.$$
To solve this, the trick is to take one more (partial) derivative and cyclically premutate the indices
$$\partial_{\rho} \partial_{\mu} \xi_{\nu} + \partial_{\rho} \partial_{\nu} \xi_{\mu} = 0 \; \Rightarrow \; \partial_{\mu} \partial_{\nu} \xi_{\rho} + \partial_{\mu} \partial_{\rho} \xi_{\nu}=0, \quad \partial_{\nu} \partial_{\rho} \xi_{\mu} + \partial_{\nu} \partial_{\mu} \xi_{\rho}=0.$$
Adding the first two of these equations and subtracting the third, finally leads to
$$\partial_{\rho} \partial_{\mu} \xi_{\nu}=0 \; \Rightarrow \; \xi_{\nu} = a_{\nu} + \Omega_{\nu \rho} x^{\rho}.$$
Plugging this again in the killing equation, yields
$$\partial_{\rho} \xi_{\nu} + \partial_{\nu} \xi_{\rho} = \Omega_{\nu \rho} + \Omega_{\rho \nu}=0 \; \Rightarrow \; \Omega_{\rho \nu} = - \Omega_{\nu \rho}.$$
Indeed, as we see the 10 independent killing fields, defined by setting all but one of the 10 independent parameters ##a_{\nu}## and ##\Omega_{\mu \nu}## to zero, indeed generate space-time translations, rotations (parametrized with the 3 independent ##\Omega_{\mu \nu}## with ##\mu,\nu \in \{1,2,3 \}##) and boosts (parametrized with the three independent ##\Omega_{\mu 0}##, ##\mu \in \{1,2,3 \}##).
