 Solution

Solving ${\Lambda }^T\eta \Lambda =\eta$ with $\det \Lambda =1$ shows that every element is either $\Lambda (\phi )$ or $-\Lambda (\phi )$. HenceEach 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.