Let $G$ be a connected Lie group with identity $e$ and let $g\in G$. Must there exist a left-invariant vector field $X_g = dL_g(v)$ (for some $v \in \mathfrak g$) such that the flow $\phi_t(e)$ of $X$ through $e$ passes through $g$? I believe this is equivalent to the question of whether the exponential map from $\mathfrak g$ to $G$ is surjective.
[Math] Flows of left-invariant vector fields on a Lie Group
differential-geometrylie-groups
Best Answer
No. The equivalence you describe holds, and it's well-known that the exponential map fails to be surjective for $\text{SL}_2(\mathbb{R})$. Any $g$ with this property necessarily has a square root, which $\left[ \begin{array}{cc} -1 & 1 \\\ 0 & -1 \end{array} \right]$ doesn't. (Since any square root in $\text{SL}_2(\mathbb{R})$ has determinant $1$, it must have eigenvalues $\pm i$, so in particular it is diagonalizable.)
The exponential map is surjective if $G$ is either compact or nilpotent.