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I'm trying to solve the following problem of the book "Grupos de Lie - Luiz A. B. San Martin":

Question: Let $G$ be a connected Lie Group with Lie algebra $\mathfrak{g}$. Suppose that $X,Y \in \mathfrak{g}$ generate $\mathfrak{g}$ (i.e. $X$ and $Y$ are not contained in any proper subalgebra of $\mathfrak{g}$, or in a equivalent way, the successive brackets of $X$ and $Y$ generates the Lie Algebra $\mathfrak{g}$). Show that the $1$-parameter groups $\text{exp}(tX)$ and $\text{exp}(tY)$ generate $G$.

Consider $ B = \langle \exp (t X), \exp (s Y)\rangle$ as the smallest group $\subset$ $G$, such that $\exp (tX)$ and $\exp (sY)$ belongs to $B$ , $\forall\ t,s \in \mathbb{R}$.


Some ideas.

Once $G$ is connected the problem is equivalent to showing that in a neighborhood $V$ of $1$ in $G$, the result holds. So I'm trying to find a smart way to generate the brackets of $X$ and $Y$, by derivating curves contained in $\langle \exp (tX), \exp (s Y)\rangle $ in $t=0$. For example, something like the curve $\exp(X) \exp(tY) \exp(-X)$, in this specific case

\begin{align*}\left.\frac{d}{dt}\exp(X) \exp(tY) \exp(-X) \right|_{t=0} &=\left.\exp(te^{ad(X)}Y) \right|_{t=0}\\ &= e^{ad(X)}Y\\ &= Y + [X,Y] + \frac{1}{2}[X,[X,Y]] + ..., \end{align*}

If I would be able to construct curves such that $\alpha_i: \mathbb{R} \to G $, such that $\text{Im}(\alpha_i)\subset \langle \exp (tX), \exp (sY)\rangle, $ $\alpha_i(0) = 1$ and $$\left\{\left.\frac{d}{dt} \alpha_1 (t) \right|_{t=0},..., \left.\frac{d}{dt} \alpha_n (t)\right|_{t=0} \right\},$$

is a basis of $\mathfrak{g}$, then the local diffeomorphism at $0$, \begin{align*} \varphi: \mathbb{R}^n &\to G\\ (t_1,...,t_n)&\mapsto \alpha_1(t_1)\cdot ...\cdot \alpha_n(t_n), \end{align*}

would solve the problem, however, this idea has not been fruitful.

Can anyone help me?

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