I'm sorry for this lexical, probably extremely elementary, question. But what is a pseudo-rotation? I just read this term for the first time, in the beginning of the 4th chapter book of CFT by Di Francesco & al. I would say it may be an hyperbolic rotation or a rotation followed by a parity operation (with determinant equals to -1). Couldn't find it on google, so it doesn't seem to be a standard terminology, otherwise please forgive my ignorance.
More generally, since a manifold $(M,g)$ equipped with a metric $g$ of signature $(p,q)$ is called a pseudo-Riemannian manifold, it is natural to call $O(p,q)$ the pseudo-orthogonal group and call $SO(p,q)$ the group of pseudo-rotations. In other words, the prefix pseudo refers here to that $p$ or $q$ are not zero.
II) Note that in the same setting of $O(d)$, $O(d-1,1)$, and $O(p,q)$, a pseudovector (and more generally a pseudotensor) use the prefix pseudo in a different way, namely to denote an additional sign flip in the transformation law under an orientation reversing transformation.
III) On page 38 in chapter 2 of the book CFT by Di Francesco et al. is written:
One may draw two conclusions:
IV) Finally, let us mention that there exists an unrelated notion of pseudorotations in chemistry.