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

I.e

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

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

I am probably being really thick, but what are the killing vectors that preserve the surfaces of 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 t-x_3=constant, where the constant could be zero I suppose if that makes it easier.