Let G be a simple algebraic group over an algebraically closed field K of characteristic p > 0, where p is a good prime for G, and let u ∈ G be a unipotent element. We study the embedding of u in abelian unipotent subgroups and certain reductive subgroups of G and use this information to obtain information on CG(u). Of particular interest are Z(CG(u)) and the reductive part of CG(u). The process of embedding a unipotent element u ∈ G in a connected, abelian, unipotent group is sometimes called saturation. In previous work [?] saturation results were achieved when |u| = p, where it was shown that there is a unique 1dimensional unipotent group, the “saturation” of u, which contains u and which is contained in a restricted (sometimes called good) A1 subgroup of G. Here an A1 subgroup of G is restricted if all weights on the adjoint module of G are at most 2p− 2. It was shown in [?] that these A1 subgroups provide the basis for a variant of the Steinberg tensor product theorem, hence the name. It was also shown in [?] that CG(u) can be factored as a product of the unipotent radical and the centralizer of a restricted A1 subgroup containing u. In this paper we extend these results to cover unipotent elements of arbitrary order. We describe a certain 1-dimensional torus which normalizes CG(u) and then decompose Z(CG(u)) into indecomposable summands, each invariant under the torus and such that u is contained in one of the summands, say W . If |u| = p > p, we show that the saturation of up , described above, coincides with the subgroup of elements of order at most p in W . We next establish the existence of a pair of reductive groups J,R such that each is the centralizer of the other, u is a semiregular unipotent element of J , and R is the reductive part of CG(u). These subgroups provide insight into a number of issues surrounding unipotent elements. For instance, the reductive part of CG(u) appears as the centralizer of a reductive group and this is helpful