Let $G$ be a simple Lie group of dimension $n$ (connected or even simply connected). Let $T$ be a maximal torus of dimension $d$. Notice that $\frac{n}{d}$ is an integer which I will denote by $m$. Let $g\_{1},\dots g\_{m}$ be $m$ elements of $G$ such that for $i\neq j$ we have
$$ g\_{i}T\cap g\_{j}T=\emptyset$$ 


Notation: $g\_{i}T=T\_{i}$.


We define a smooth map. 
$$V:T\_{1}\times \dots \times T\_{m}\rightarrow G$$
$$(x\_{1},\dots ,x\_{m})\mapsto x\_{1}x\_{2}\dots x\_{m}$$


My question is the following:
is the map $V$ a local diffeomorphism around the point $(g\_{1},\dots ,g\_{m})$ i.e.
the linear map (derivation) $T\_{(g\_{1},\dots ,g\_{m})}V$ is of rank $n$ ?



Local diffeomorphism from a torus to a Lie group
No. Take $g\_2=1$, $g\_1\notin T$, then $g\_1 T\cap g\_2 T=g\_1T\cap T=\emptyset$.
The differential of the map
$$\phi\colon T\times T\to G,\quad (t\_1,t\_2)\mapsto g\_1 t\_1 g\_2 t\_2=g\_1t\_1t\_2$$
at the point $(1,1)\in T\times T$ (or at any other point of $T\times T$) is clearly of rank $d=\mathrm{dim}\, T$, and not $2d$, which implies the answer NO to the original question.


