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,
- Q1) If $G=AB$ is a factorization ($G=AB;\ A\cap B=\{1_G\}$), do we have $Lie(A)\oplus Lie(B)=Lie(G)$ ?
- Q2) Does the result hold if we just have $G=AB$ (and still $Lie(A)\cap Lie(B)=\{0\}$) without supposing that $A\cap B=\{1_G\}$ ?
- Q3) What are the references about these questions ?