Problem
Problem 4.37 Prove that the following are Lie groups:
(i) Each finite-dimensional real vector space with its structure of additive group. In particular $\mathbb{R}^{n}$. (ii) The set of non-zero complex numbers $\mathbb{C}^{*}$ with the multiplication of complex numbers.
(iii) $G \times H$, where $G, H$ are Lie groups, with the product $(g, h)\left(g^{\prime}, h^{\prime}\right)=$ $\left(g g^{\prime}, h h^{\prime}\right), g, g^{\prime} \in G, h, h^{\prime} \in H$. In general, if $G_{i}, i=1, \ldots, n$, is a Lie group, then $G_{1} \times \cdots \times G_{n}$ is a Lie group.
(iv) $T^{n}$ for $n \geqslant 1$ (toral group).
(v) Aut $V$, where $V$ is a vector space of finite dimension over $\mathbb{R}$ or $\mathbb{C}$, with the composition product, and in particular $\operatorname{GL}(n, \mathbb{R})=\operatorname{Aut}_{\mathbb{R}} \mathbb{R}^{n}$ and $\operatorname{GL}(n, \mathbb{C})=$ Aut $_{\mathbb{C}} \mathbb{C}^{n}$
(vi) $K=\mathbb{R}^{n} \times \operatorname{GL}(n, \mathbb{R}), n>1$, with the group structure defined by
$$
(x, A)\left(x^{\prime}, A^{\prime}\right)=\left(x+A x^{\prime}, A A^{\prime}\right)
$$
The relevant theory is developed, for instance, in Warner [13].