This is all very classical and dealt with in numerous books on Lie algebras. For instance, analyzing finite dimensional representations depends on complete reducibility and the (easy) explicit construction of all irreducibles. The dependence of *h* and *f* on *e* (up to certain conjugacy conditions) is also standard in treatments of $\mathfrak{sl}_2$ embeddings. The question isn't really about "linear algebra" as tagged, but rather belongs to the general theory of finite dimensional representations of semisimple Lie algebras over fields like $\mathbb{C}$. There is no extra benefit in treating it just as a matrix computation problem, especially because *E* doesn't strictly determine the choice of *F* and *H* in matrix terms.

ADDED: I still don't understand what is actually being asked here, I guess. The theory of nilpotent elements in a (complex) semisimple Lie algebra, including $\mathfrak{sl}_2$ triples and their finite dimensional representations, is well understood.

On the other hand, if one starts with a (nonzero) nilpotent matrix *E*, there are many possible choices of a matrix *F* and resulting semisimple matrix *H*. Concretely, one could transform *E* by similarity to its rational canonical form (computable if given enough time), then get an *F* from the transpose by applying the inverse similarity. As noted in some answers, conjugating *F* by a matrix which commutes with *E* would give a possibly new choice. Why would one want to do all of this? Note that a resulting $\mathfrak{sl}_2$ representation would typically be reducible. And at the outset, it's hard to recognize that a large matrix *E* is truly nilpotent, due to possible roundoff errors, etc.