I apologize if this has been asked before, and would appreciate a reference.
Let $G$ be a real Lie group (if it matters, I'm satisfied with $G$ being a matrix Lie group, say $SL_n(\Bbb{R})$). Suppose that the Lie algebra of $G$ can be written as a direct sum $\mathfrak{g} = V \oplus W$ of vector subspaces. Is it true that there exists an open neighborhood of the identity that is contained in $\exp(V)\exp(W)$? In other words, can I always find an open ball $B_r(e) \subset G$ of identity so that every element in the ball can be written as $\exp(v) \exp(w)$ for some $v \in V$ and $w \in W$?
If not, is it true if in addition, $W$ is a Lie subalgebra of $\mathfrak{g}$? If not, is it true if in addition, $W$ is a nilpotent Lie subalgebra of $\mathfrak{g}$?
Again I'm okay with a solution for $G = SL_n(\Bbb{R})$, though a proof for the general case would be wonderful. Thanks a lot.
Define $\psi: V \oplus W\cong V \times W\to G$ by $\psi(X,Y)=\exp(X)\exp(Y)=\rho(\exp(X),\exp(Y))=\rho(\eta(X,Y))$, where $\eta(X,Y)=(\exp(X),\exp(Y))$. Then, $\psi$ is differentiable and \begin{eqnarray} d(\psi)_{(0,0)}&=&[d(R_{\exp(0)})_{\exp(0)} \quad d(L_{\exp(0)})_{\exp(0)}]=\operatorname{Id}. \end{eqnarray}
Therefore, from the Inverse Function Theorem, there exists open nbhd's $U_1 \subset V$, $U_2 \subset W$ and $G'\subset G$ a neighborhood of $1 \in G$ such that $\psi:U_1 \times U_2 \to G'$ is a diffeomorphism.