If $\mathfrak g\subseteq \mathfrak{gl} (V)$ is a finite dimensional Lie algebra, then we have a usual Jordan decomposition in $\mathfrak{gl} (V)$, $$ \forall x \in \mathfrak{g}\ \exists x_s, x_n\in \mathfrak{gl}(V),\quad x = x_s + x_n, \ x_s \text{ is diagonalizable}, x_n \text{ is nilpotent} $$ suppose $\dim V = n$, and $x$ is in Jordan canonical form for a given basis of $V$. Then we can choose a basis of $\mathfrak{gl} (V)$ as $\{e_{ij}\}$ whose $(i,j)$ entry is $1$ and all other entries are vanishing. By this basis, we can easily check $\mathrm{ad}_{x_s}$ is diagonalized and $\mathrm{ad}_{x_n}$ is nilpotent. Since the uniqueness of Jordan decomposition for $\mathrm{ad}_x$, usual Jordan decomposition is preserved by adjoint representation.
In general cases, if any finite dimensional representation preserves the usual Jordan decomposition?
Maybe only consider the case that vector spaces are over some algebraic closed field.
And I have seen a theorem states that (I am not really sure), there exist polynomials $S,N$ without constant term such that $x_s = S(x), x_n = N(x)$ and they only depend on the Lie algebra itself.
Thanks a lot!
No, Jordan decomposition is not preserved in general. For example, take a one-dimensional Lie subalgebra generated by any matrix: the matrix may be semi-simple or nilpotent (or anything in between), but the abstract Lie algebra is always the same (abelian).
The most general results here are the following: for $\mathfrak{g}$ a semi-simple Lie algebra over the complex numbers, the adjoint representation is injective and we define the Jordan decomposition by $$\mathrm{ad}(x)=\mathrm{ad}(x_s)+\mathrm{ad}(x_n)$$ so that $\mathrm{ad}(x_s)$ and $\mathrm{ad}(x_n)$ are the semi-simple and nilpotent parts of $\mathrm{ad}(x)$ (one must check this is well-defined).
Theorem: Let $\mathfrak{g} \subseteq \mathfrak{gl}(V)$ by a semi-simple Lie subalgebra. Then $\mathfrak{g}$ contains the semi-simple and nilpotent parts of each $x \in \mathfrak{g}$, and these are $x_s$ and $x_n$.
The theorem implies that the images by any representation of $x_s$ and $x_n$ are the semi-simple and nilpotent parts of the image of $x$. A frequently used reference for this is Humphrey's book Introduction to Lie algebras and representation theory. See the Theorem from 6.4.