Consider the Lie algebra $sl_2$ with the standard basis $(e,f,h),$ where \begin{equation*}\label{sl2} [h,e]=2\,e, [h,f]=-2\,f,[e,f]=h. \end{equation*}
Let $V$ be finite-dimensional $sl_2$-module and let we know that element $e$ acts on $V$ as linear operator with a matrix $E$ (in some fixed basis).
Question. How one may find the matrix $F,H$ which correspond to elements $f$ and $h.?$
It is posible to recover $F,H$ from $E?$
Edited in light of clarifications made by OP:
Given a nilpotent matrix $E$ acting on a finite dimensional vector space $V$, it is always possible to extend it to a representation of $sl_2$ in such a way that it represents $e$. The extension is almost never unique: conjugating the representing matrices $F$ and $H$ by anything in the centralizer of $E$ gives a new extension.
The existence statement is the Jacobson-Morozov lemma (part of whose standard proof is reproduced in another answer) applied to the semisimple Lie algebra $sl(V)$. See Proposition 2 of section 2 of paragraph 11 of Bourbaki's "Lie Groups and Lie Algebras", Chapter VIII (see the Corollary following the Proposition for the extent to which uniqueness is true: basically, up to conjugacy).
On the other hand, if you have some additional rigid structure, there might be a unique extension. For instance, if you know a contravariant form and have an orthonormal basis at your disposal then $F$ is the transpose of $E$ (written in terms of the given orthonormal basis) and $H$ is determined by $H=[E,F]$.
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.
$F$.
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Commented
Nov 9, 2010 at 15:35
The element $E$ is nilpotent; if the representation is irreducible, it has the maximal possible rank. Denote the rank $k$ so that the dimension of the underlying vector space $V$ will be $k+1$. There is a basis $(v_0,\ldots,v_k)$ of $V$ such that
$F$ will be a Jordan cell i.e. $Fv_i=v_{i-1}, Fv_0=0$. $v_{k}$ is the highest weight vector.
$Hv_i=(2i-k)v_i$.
$Ev_i=i(k-i+1)v_{i+1},Ev_k=0$.
These formulae in some form or another can be found in many sources (e.g in Serre's Lie algebra book).
[upd: one can reconstruct all this from $F$ given in some basis by setting $v_0$ to be any nonzero vector in $Im F^k=\ker F$ and then defining $v_1,\ldots, v_k$ by formula 3 above. With this definition formulae 1 and 2 will be true as well.]
For every irreducible finite-dimensional representation there's a basis in which the matrices take particularly simple form (this should be in most of the introductory representation theory books). You can cook up you matrices F,H from there by an appropriate change of basis.
