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, 
><ol>
<li>Providing $G=AB$ 
<li> and if it is a factorization (means that 
the decomposition is unique or, equivalently, 
$A\cap B=\{1_G\}$)
</ol>
then $Lie(A)\cap Lie(B)=\{0\}$
 
 
My questions are the following 
  
><ul>
<li>Q1) If $G=AB$ is a factorization ($G=AB;\ A\cap B=\{1_G\}$),  
 do we have 
$Lie(A)\oplus Lie(B)=Lie(G)$ ?
<li> Q2) Does the result hold if we just have 
$G=AB$ (and still $Lie(A)\cap Lie(B)=\{0\}$)  
without supposing that $A\cap B=\{1_G\}$ ?
<li> Q3) What are the references about these 
questions ? 
</ul>