Let $G$ be a Lie group over the reals with Lie algebra $\mathfrak{g}$ and let $B = \{v_1, \cdots, v_n\}$ be a basis of $\mathfrak{g}$. Consider the map
$$f : \mathbb{R}^n \to G \quad (t_1,\cdots,t_n) \mapsto exp(t_1 v_1)\cdots exp(t_n v_n)$$
Here is my question: can we impose some conditions on $G$ and the basis $B$ to make $f$ a diffeomorphism? And what if we change $f$ by $$f((t_1,\cdots ,t_n)) = exp(t_1v_1 + \cdots t_nv_n)$$
Thanks for your help!
$\textbf{EDIT}$:
Perhaps I should have mentioned how I came up with the first question. I was reading a paper by M.Ratner $\textit{Strict measure rigidity for unipotent subgroups of solvable groups}$ and at the begginning of the proof of Proposition $2.1$, the author made a similar statement to my first question.
In Ratner's article, $G$ is assumed to be connected, simply connected and nilpotent. $f$ is as above and the basis $B = \{b_1, \cdots, b_n\}$ is assumed to be regular. The arthor claimed $f$ is then a diffeomorphism.
Definition of regularity: fix a basis $B = \{b_1, \cdots, b_n\}$ of $\mathfrak{g}$, we can write $v = \sum \alpha_{i}(v) b_i$ for any $v\in \mathfrak{g}$. The basis $B$ is called $\textbf{triangular}$ if $$\alpha_{k} ([b_i, b_j]) = 0, \quad \forall k\le max\{i,j\}$$ And the basis $B = {b_1, \cdots, b_n}$ is called $\textbf{regular}$ if it is a permutation of triangular basis.
I can't prove the case in that article, so I wonder if there exists some general criteria, and asked the second question as a analog. Thanks to @Callum, the second question is thoroughly dealt with, and I wonder if someone could help to prove the first question under the setting of Ratner. Of course, comments on the general case are also higly appreciated!
Here is Ratner's paper if you need:https://mathscinet.ams.org/mathscinet/article?mr=1062971
