Is Spherically Symmetric Spacetime Algebraically Special?

[afs]

Hi,
I've been reading about Petrov classification and I have a question (in fact this is an exercise from Wald's General Relativity): How can we prove that spherically symmetric spacetimes are algebraically special, using the fact that the Weyl tensor, as the principal null directions are invariant over isometries? I've look over the internet, but I coudn't find a clue.
Thanks for any help!

 

[afs]

I just got one insight for this question, please report me if I am wrong. Spherically symmetric spacetimes possesses a smooth two-parameter group of isometries, and a single vector can be transformed in any other, "rotated" one. As this smooth-parametrized isometries leave the null directions invariant, the two-dimensional region of the tangent space generated by the application of theis group of isometries corresponds to a single null direction. Therefore at least two directions must coincide, and the spacetime is algebraically special.