Indefinite orthogonal group

Example

Let $p, q \ge 0$ be integers with $n = p + q \ge 1$, and let $$\eta = \begin{pmatrix} I_p & 0 \\ 0 & -I_q \end{pmatrix}.$$ The matrix $\eta$ defines the quadratic form $Q(x) = x^T \eta x = x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_n^2$ on $\R^n$, of signature $(p, q)$. The indefinite orthogonal group is $$O(p, q) = \set{A \in GL_n(\R) : A^T \eta A = \eta},$$ the group of linear maps of $\R^n$ that preserve $Q$. It is a closed subgroup of $GL_n(\R)$, so it is a Lie group.

Check with $p = q = 1$: for every $t \in \R$, the matrix $$A_t = \begin{pmatrix} \cosh t & \sinh t \\ \sinh t & \cosh t \end{pmatrix}$$ satisfies $A_t^T \eta A_t = \begin{pmatrix} \cosh^2 t - \sinh^2 t & 0 \\ 0 & \sinh^2 t - \cosh^2 t \end{pmatrix} = \eta$, so $A_t \in O(1, 1)$. Its entries are unbounded as $t \to \infty$, so $O(1, 1)$ is not compact.