Local diffeomorphism from a torus to a Lie group

Question

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 $n$ ?

Answer 1

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.