Geometrically simple quasi-abelian varieties

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 $\mathbb{Q}$.