Toroidal algebraic groups

Many of the more striking elementary properties of abelian varieties generalize to other kinds of algebraic groups, e.g. tori, i.e. direct products of multiplicative groups Gm, and in fact to extensions of tori by abelian varieties. This seems to have been more or less known for some time, but explicit statements are hard to find, a lack we attempt to remedy here. We may mention specifically a recent paper of S. Arima [I1], reading between the lines of which might lead one in this direction. In what follows, the better-known structural facts on algebraic groups are used freely; most of these are in [2] and [4], while the relevant facts on abelian varieties can be found in either [7] or [3].