If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ is a Cartan-decomposition, then the restriction of $\exp$ to $\mathfrak{p}$ is a diffeomorphism onto its closed image. An example thereof is $G=SL_n(\mathbb{R})$, where $\mathfrak{k}$ are the skew-symmetric matrices $\mathfrak{p}$ are the symmetric traceless matrices. Then $\exp$ is certainly injective on a neighbourhood of the form $U\times V$ with $U\subseteq\mathfrak{k}$ and $V\subseteq\mathfrak{p}$ open.
My Question is now: does there exist some open $W\subseteq\mathfrak{k}$ such that $\exp$ is injective on $W\times \mathfrak{p}$? From the way the Cartan decomposition is built I would expect that one can achieve injectivity on an open neighbourhood of $0\times\mathfrak{p}$ (although I cannot write down a precise argument for this at the moment), so the question is whether this open neighbourhood is globally of product type.
