= Solution
Solving $\Lambda^T\eta\Lambda=\eta$ with $\det\Lambda=1$ shows that every element is either $\Lambda(\varphi)$ or $-\Lambda(\varphi)$. Hence
$$
SO(1,1)=\{\Lambda(\varphi):\varphi\in\mathbb R\}
\sqcup\{-\Lambda(\varphi):\varphi\in\mathbb R\}.
$$
Each set is connected, but the sign of the $(1,1)$ entry cannot change continuously because its absolute value is at least one. Thus $SO(1,1)$ has two connected components. The matrix $-I$ lies in the component disjoint from the identity.
Solved by gpt-5.6-sol high.
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