Hi to all, 


Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n\_{\nu}$ ($\mathbb{R}^n\_{\nu}$ is the semi-Euclidean space) and $G\_1$ ,$G\_2$ two Lie subgroups of $G$. Let $G\_1 \times G\_2$ as a Lie subgroup of $G$ by identifying it with the inner product of $G\_1$ and $G\_2$ (suppose $G\_1$ , $G\_2$ have the required properties to form inner direct product) and let $G\_1 \times G\_2$ acts on $\mathbb{R}^n\_{\nu}$ by the induced action from $G$ . What can we say about the relation between the orbits of $G\_1 \times G\_2$ and the orbits of $G\_1$ and $G\_2$? Is a $(G\_1 \times G\_2)$-orbit isometric (or at least diffeomorphic) to the product of the $G\_1$-orbit and $G\_2$-orbit? 


Best for you.



Orbits of Product Lie Groups Action
If all actions are proper, orbits are closed embedded submanifolds.
If not, then they are only initial submanifolds (unique injectively immersed submanifolds with a possibly finer topology - think of a 1-param. subgroup of the torus with irrational slope;
see thm 6.4 in [(here)](http://www.mat.univie.ac.at/~michor/dgbook.pdf))


Now the $(G\times G\_1)$-orbit through $x$ is diffeomorphic to $(G\_1\times G\_2)/(G\_1\times G\_2)\_x$ (mod out the isotropy group). 
If $(G\_1\times G\_2)\_x = (G\_1)\_x\times (G\_2)\_x$
then the orbit is a product. We always have 
$(G\_1\times G\_2)\_x \supset (G\_1)\_x\times (G\_2)\_x$
so the $(G\times G\_1)$-orbit through $x$ is always a submersive quotient of the product of the two orbits.


The source I cited has more information.


