"Classification" can mean more than one thing, but it's useful to be aware of the extensive development of adjoint quotients by Kostant, Steinberg, Springer, Slodowy, and others. This makes sense for all semisimple (or reductive) groups and their Lie algebras over any algebraically closed field, perhaps avoiding a few small prime characteristics. Older sources include Steinberg's 1965 IHES paper on regular elements (MSN and article) and the Springer–Steinberg portion of the
1970 Lecture Notes in Math. vol. 131 (MSN and chapter). (For an overview with references, based partly on Steinberg's Tata lectures, see Chapter 3 of my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups (MSN, book, and chapter).) While the general focus has been on developing a picture of the collection of all classes or orbits as some kind of "quotient", quite a few special features of the classical types are also brought out in the Springer-Steinberg notes. As suggested by Victor, Roger Carter's book Finite Groups of Lie Type (MSN) also has a lot of related material but with special emphasis on nilpotent orbits. The Jordan decomposition does reduce many classification questions to the nilpotent case, at least in principle, if you are willing to deal with various centralizers along the way.
[ADDED] The paper in J. Math. Physics linked below gives a nice concrete answer to the original question, building on some of the older theory but using mainly tools from linear algebra and basic group theory. This is the traditional approach of most physicists, though papers in this mixed journal are sometimes unreliable and contain mathematics of the sort probably not usable in physics but also not publishable in math journals. Djokovic and his collaborators are more reliable than most, fortunately, and he has written many papers using parts of Lie theory as well. One downside is the narrower perspective than found in the notes of Springer–Steinberg, for instance. But it all depends on whether you want to work over other fields or want to organize the classes/orbits more conceptually.
Here is a MathSciNet reference:
MR708648 (85g:15018) 15A21 (17B99 17C99)
Djokovic´,D. Zˇ . [¯Dokovic´, Dragomir Zˇ .] (3-WTRL); Patera, J. [Patera, Jirˇ´ı] (3-MTRL-R);
Winternitz, P. (3-MTRL-R); Zassenhaus, H. (1-OHS)
Normal forms of elements of classical real and complex Lie and Jordan algebras.
J. Math. Phys. 24 (1983), no. 6, 1363–1374 (review by R.C. King).
(MSN and article.) Their references include the work by Burgoyne–Cushman, Milnor, Springer-Steinberg mentioned by Bruce and me.
Here are a few extended comments. First, it's always desirable to re-examine basic material as more of it accumulates and makes the research frontier look impossibly remote to students. The handy book What Every Young Mathematician Should Know (But Didn't Learn in Kindergarten) gets harder to write every year.
It's never easy to say what ingenious new approaches are possible, but the question in the header has already been tackled (unsuccessfully) in a moderately paced textbook by Karin Erdmann and her former student Mark Wildon Introduction to Lie Algebras (Springer 2006, reprinted with some corrections in softcover format). The authors maintain a list of corrections to both versions, including a further correction to Theorem 9.16 on preservation of Jordan decomposition. They tried to avoid Weyl's theorem on complete reducibility but tripped over a hidden obstacle.
As this cautionary example indicates, it is tempting to simplify proofs but not always easy. The rigorous approach taken by Bourbaki (and Serre) to such matters is reliable though not always user-friendly. In any case, semisimple Lie algebras can be studied over an algebraically closed field of characeristic 0 using just some linear algebra and basic abstract algebra. It's not at all necessary, except for motivation, to deal with complex coefficients, Lie groups, or linear algebraic groups. Even so, there are sophisticated arguments including Cartan's criterion for solvability which seem hard to avoid.
Historically the classification of simple Lie algebras over a field such as $\mathbb{C}$ doesn't involve notions like Chevalley-Jordan decomposition or Weyl's complete reduciblity theorem. It's at least partly a matter of taste which approach to take, but Casimir operators and such do come up naturally if one gets into representation theory. My main concern about restructuring the foundations is that it should be done rigorously and in a way that doesn't force students to go back and start over again if they decide to pursue the subject further.
P.S. Some other relevant questions on MO can be found by searching "Jordan decomposition".
Best Answer
Any finite dimensional simple Lie algebra over an algebraically closed field of characteristic $p>3$ contains a nonzero element $x$ such that $ad(x)$ is semisimple. This is a nontrivial fact, and the only proof I can think of relies on the classification. For $p>3$, the simple Lie algebras split into three families: Lie algebras of simple algebraic groups and their quotients, filtered Lie algebras of Cartan type, and the Melikian algebras (which only occur when $p=5$). The first class is easy to sort out, of course. The third class has a very explicit presentation which is easy to find in the literature. Then one can see straight away that each Melikian algebra $\mathcal{M}(m,n)$ admits two commuting linearly independent elements $t_1,t_2$ sich that $ad(t_i)^p=ad(t_i)$ for $i=1,2$.
The real issue with this problem (as well as with many similar problems) is the existence of filtered Lie algebras of Cartan type $H$ corresponding to Hamiltonian forms of the first kind. Such Hamiltonian forms involve parameters which are sometimes organised as sets of matrices in canonical Jordan form. Sadly, the commutator relations in the corresponding Lie algebras depend on these parameters as well, which makes computations an unpleasant experience.
However, the good news is that any finite dimensional simple Cartan type Lie algebra $L$ contains a unique maximal subalgebra of smallest codimension. It is called the $standard\ maximal\ subalgebra$ of $L$ and often denoted $L_{(0)}$. Skryabin proved in [Comm. Algebra, 23 (1995), 1403-1453] that under our assumptions on $p$ the subalgebra $ad(L_{(0)})$ of $\mathfrak{gl}(L)$ is closed under taking $p$-th powers; see Theorem 2.1 in $loc.\ cit$. By maximality, $L_{(0)}$ does not consist of $ad$-nilpotent elements. Since $ad(L_{(0)})$ is closed under taking $p$-th powers, it is straightforward to see that there is a nonzero element $x\in L_{(0)}$ such that $ad(x)$ is semisimple (in fact, one can say a lot more than that).
I have no idea as to what happens when $p$ is $2$ or $3$ as we have no classification in these characteristics. This adds to a long list of open problems of which my favourite is: does there exist a finite dimensional simple Lie algebra $L$ with a finite automorphism group. This never happens when $p>3$ and there is a rather short proof of this fact in the literature.