The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for elliptic functions.
To "classify" addition theorems it is necessary to put some restriction on the type of function G admitted, such that In this identity one can assume that F and G are vector-valued (have several components).
An algebraic addition theorem is one in which G can be taken to be a vector of polynomials, in some set of variables.
The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law.
Therefore, the old conclusions about the scope of global algebraic addition theorems can be said to hold.