A Lie Group and Lie Algebra Problem I am a physicist doing some research and I come across with the following Lie algebra problem.
Consider the Lie Group $G$ (compact and connected, if you wish), and two generators in the corresponding Lie algebra $X$ and $Y$. By successive action of exponential map you can get the following element in the Lie group
$$e^{\alpha_1 X}e^{\beta_1 Y}...e^{\alpha_n X}e^{\beta_n Y} \in G.  $$
The question is: when will the whole Lie group be generated by the action above? And if some part of the Lie group cannot be generated, what is the subgroup that can be generated?
Extensions: What is the closure of the generated subgroup? Can you extend the above results to 3 or more generators?
Example: 
(1) Consider $SU(2)$ and $X=i\sigma_x, Y=i\sigma_y$ ($\sigma$ are Pauli matrices), then they can generate the whole $SU(2)$.
(2) Consider $S^1\times S^1$ as a Lie group and $X=Y=i(a,b)$ (in the naturally chosen coordinate system). Of course only a one dimensional subgroup can be generated. However, if $\frac{a}{b}$ is irrational, the closure is the whole group.
Thank you for your attention! 

Nice question. For $SU(2)$ or $SO(3)$ you get I think the Euler's angles, and there are analogous results for $SU(N)$. I don't know about the general case, it does look plausible.
Certainly the closure of the group generated by those exponential is the connected component of $e$. Either you use Cartan's theorem on closed subgroup, or use basic facts like 
$$\exp(X+Y) =  \lim_{n\to\infty} (\exp\frac{1}{n}X\exp\frac{1}{n}Y)^n\\
\exp[X,Y]= \lim_{n\to\infty} (\exp \frac{1}{n}X\exp\frac{1}{n}Y\exp(-\frac{1}{n}X) \exp(-\frac{1}{n}Y) )^{n^2}$$
