I believe it should be $-A(X)$ instead of $A(Y)$ in the first row. I ran the computations and the result followed.

Thank you! I tried to use the same result, but couldn't formulate it as cleanly, and got lost somewhere in the index notation.

You are right about the index, but this was a typo only in the post (I edited it). Otherwise, the system is obviously trivial once we know that $X$ is Killing. I assume the idea is to first arrive at the curvature condition to illustrate what the extreme case is, and apply the symmetry $X_{ij} = - X_{ji}$ afterwards.