In Chapter 4, we saw that the algebra R(G) of representative functions on the analytic group G could be decomposed as a group ring of the abelian group Q = exp(Hom(G, ℂ)) with coefficients from any left algebraic group structure A in R(G). We saw further that left algebraic group structures arise from and give rise to split hulls of G. Thus, the study of the possible decompositions R(G) = A[Q] is equivalent to the search for the split hulls of G. (We will restrict our attention to basal left algebraic group structures and their corresponding split hulls.) In Chapter 3, we saw how to construct a split hull of an analytic group; a review of that construction will reveal that the only choices made were the selection of a nucleus and essentially the choice of a Cartan subgroup of that nucleus (a Cartan subgroup is an analytic subgroup whose Lie algebra is a Cartan subalgebra). Moreover, that construction meets our criterion (4.19) so that the corresponding left algebraic group structure is basal. This suggests that if we want to reverse our construction, we begin with a basal left algebraic group structure, then pass to the associated split hull, and then we must produce a nucleus and a Cartan subgroup of it. That program turns out to be possible, and we carry it out in this chapter.

There are some additional consequences: the identification of nuclei and their Cartan subgroups can be done strictly in Lie algebra terms, once the Lie algebra of a maximal reductive subgroup is designated.