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Theorem 1.7 of Schottenloher's "Mathematical Intro to CFT" says that every conformal Killing field on a connected open subset $M$ of $\mathbb{R}^{p,q}$ for $n=p+q>2$ is of the form $$X^{\mu}(x) = 2 \langle x, b \rangle x^{\mu} - \langle x, x \rangle b^{\mu} + \lambda x^{\mu} + c + (\omega x)^{\mu}$$ where $b,c \in \mathbb{R}^{n}$, $\lambda \in \mathbb{R}$, and $\omega \in \mathfrak{o}(p,q)$. I'm trying to solve Exercise 1.8, which says that this collection is closed under the Lie bracket. To start, setting $\lambda = c = \omega = 0$ and taking the commutator with $$Y^{\mu}(x) = 2 \langle x, \tilde{b} \rangle x^{\mu} - \langle x, x \rangle \tilde{b}^{\mu}$$ for some other $\tilde{b} \in \mathbb{R}^n$, I seem to get a result with terms that are cubic in $x$ ... and this doesn't agree with the desired form above. Can anyone spot the error I'm making? If so, I'd greatly appreciate it!

[For those who are curious, the end result should be that these vector fields form a Lie algebra isomorphic to $\mathfrak{o}(p+1,q+1)$.]

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