TITLE: Decomposition theorem for semi-abelian varieties
QUESTION [5 upvotes]: Fact : 

Let $B$ an abelian subvariety of an abelian variety over a field $K$. We know that there exist an abelian subvariety $C$ of $A$ such that the restriction of addition gives an isogeny  $B\times_K C\to A$.
The analogous result is true for an linear torus.

Questions :
1)Is this result is true in the category of semiabelians varieties ?
2)Is there a good reference on semi-abelians varieties ?
Thanks !

REPLY [5 votes]: No. Take an extension of an elliptic curve by a torus which is of infinite order in the Ext group and take B the torus. (Reference: Serre, Groupes algébriques et corps de classes, Ch VII).