Problem
Problem 4.70 Let $G$ be the group defined by
$$
G=\left\{A \in \operatorname{GL}(2, \mathbb{R}): A^{t} A=r^{2} I, r>0, \operatorname{det} A>0\right\} .
$$
(i) Find the explicit expression of the elements of $G$. (ii) Find its Lie algebra.
(iii) Calculate the adjoint representation of $G$.