Killing vectors and isometry

Question

Let $X=xpartial_{t}+tpartial_{x}$ and $Y=ypartial_{t}+tpartial_{y}$ be Killing vectors on Minkowski $(-,+,+,+)$. It can be shown that $[X,Y]$ is also Killing. I get the following:
begin{equation}
[X,Y]=(x-y)partial_{t}+xpartial_{y}.
end{equation}
What isometry is this Killing associated with?

Cosmas Zachos · Answer

You did not evaluate you commutator correctly. Your two boosts commute to a rotation on the x,y plane,
$$[X,Y]=  [xpartial_{t}+tpartial_{x},  ypartial_{t}+tpartial_{y}]=xpartial_y-ypartial_x,$$
and  all three annihilate $x^2+y^2+z^2-t^2$.

Killing vectors and isometry

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