Solution
= Solution
Put $\Lambda(t)=I+tX+O(t^2)$ in $\Lambda^T\eta\Lambda=\eta$. The coefficient of $t$ is $X^T\eta+\eta X$, so it must vanish. For $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ this condition is
$$
\begin{pmatrix}2a&b-c\\b-c&-2d\end{pmatrix}=0.
$$
Hence $a=d=0$ and $b=c$. Thus the <Lie algebra> is one-dimensional with basis
$$
K=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad \mathfrak{so}(1,1)=\mathbb RK.
$$