  We define the geometric simpleness for toroidal groups. We give an example of
quasi-abelian variety which is geometrically simple, but not simple. We show
that any quasi-abelian variety is isogenous to a product of geometrically
simple quasi-abelian varieties. We also show that the ${\mathbb Q}$-extension
of the ring of all endomorphisms of a geometrically simple quasi-abelian
variety is a division algebra over ${\mathbb Q}$.

Complex Variables