The underlying question here is of interest, though I'm not sure how explicitly it has been answered in the literature. However, your header is misleading: the lemma/theorem of Jacobson-Morozov just concerns the embedding of (nonzero) nilpotent elements of a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ into copies of the unique simple 3-dimensional Lie algebra $\mathfrak{sl}_2$. (The history of the independent work done by Jacobson and Morozov is rather complicated, but anyway there is a modern treatment in N. Bourbaki, *Groupes et algebres de Lie*, Chap. VIII, $\S11$, among other places. Work of Kostant and others refined the ideas further.)

It then follows from Jacobson-Morozov along with some basic Lie theory that a corresponding Lie group homomorphism $\phi:\mathrm{SL}_2 \rightarrow G$ exists, as in the question. Here $u$ is a nontrivial unipotent element obtained from the given nilpotent element in $\mathfrak{g} = \mathrm{Lie}(G)$ by something like exponentiation. Your problem is to distinguish between embeddings of $\mathrm{SL}_2$ and $\mathrm{PSL}_2$ (equal to $\mathrm{PGL}_2$ over an algebraically closed field).

Actually, one only needs an algebraically closed field of characteristic 0 here, since semisimple Lie groups are the same as semisimple algebraic groups, while there has been careful study of prime characteristic analogues by many people to carry over some of the ideas. Moreover, the study of simple algebraic groups or Lie algebras is usually sufficient in characteristic 0.

The group $G$ has a faithful (rational) representation, though for $G$ of type $E_8$ (a useful test case since the abstract group is simple) the smallest dimension possible is 248 (adjoint representation). In all characteristics the Jordan block decomposition of each unipotent element in such a representation for each exceptional Lie type has been heroically worked out by R. Lawther in *Comm. Algebra* 23 (1995), 4125-4156. Since there 69 nontrivial unipotent classes for type $E_8$, the resulting tables are nontrivial to compute. Typically there are some Jordan blocks of *odd* size (or 1), but these seem to be compatible with representations of $\mathrm{SL}_2$, which has irreducible representations of all possible dimensions $\geq 1$. It's true that if some Jordan block is of even size, this is the only possible type of subgroup.