Wikipedia says in the Section on Topology of indefinite orthogonal group $O(p, q)$ https://en.wikipedia.org/wiki/Indefinite_orthogonal_group#Topology
Can some folks clarify the statement, perhaps there is some mistake?
In particular, it says that "The special orthogonal group has components $$\pi_0(SO(p, q)) = {(1, 1), (−1, −1)}$$ which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.
I find $(−1, −1)$ may not be the $\mathbb Z_2$-generator switching between the disconnected sectors, unless $p+q$ satisfy certain constrain.
Is Wikipedia wrong or imprecise?