in coordinates $\displaystyle (t,x_1,x_2,x_3)$ the infinitesimal isometries of minkowski space are 3 boosts and 3 rotations (and 4 displacements).


$\displaystyle (t',x'_1,x'_2,x'_3)^\mu=(t,x_1,x_2,x_3)^\mu+\epsil on\xi^\mu$

where $\displaystyle \xi$ are killing vectors given by $\displaystyle (t,x_1,0,0)$, $\displaystyle (x_2,0,t,0)$, $\displaystyle (x_3,0,0,t)$,$\displaystyle (0,-x_2,x_1,0)$, $\displaystyle (0,-x_3,0,x_1)$ and $\displaystyle (0,0,-x_3,x_2)$

I am probably being really thick, but what are the killing vectors that preserve the surfaces of $\displaystyle t-x_3=constant$ I am totally confused, or atleast how do I work them out? I expect there should be three such vectors, and are linear combinations of the above vectors.

If these where constant t surfaces, then this would be easy, it's just 3 rotations, but i need to know $\displaystyle t-x_3=constant$, where the constant could be zero I suppose if that makes it easier.