Let's recall that, a group $G$ being given, two subgroups $A,B\subset G$ are called permutable iff $AB=BA$ for the Minkowski law. It is straightforward to see that $(A,B)$ are permutable iff $AB$ is a subgroup of $G$.

Let now $G$ be a finite dimensional Lie group (real to begin with) and suppose that if $A,B\subset G$ are Lie subgroups (then closed by Cartan's theorem), one can easily show that, providing $G=AB$ and if it is a factorization (means that the decomposition is unique or, equivalently, $A\cap B=\{1_G\}$), then $Lie(A)\cap Lie(B)=\{0\}$

My questions are the following