Let $V$ be a finite-dimensional vector space over a perfect field (e.g., a characteristic zero field). The (additive) Jordan decomposition of an endomorphism $g \in \mathrm{End}(V)$ says that you can write $g$ as
$$
g = g_s + g_n \ ,
$$
where $g_s$ is semisimple and $g_n$ is nilpotent. The Encyclopedia of Mathematics adds that this decomposition can be generalized to locally finite endomorphisms of infinite-dimensional vector spaces.
This generalization seems quite elementary and straightforward, but I don't know any other reference where this result is developed. So I would be grateful if anyone can provide some.
Best Answer
There are several references about this generalisation of Jordan-Chevalley decompositions arising in the theory of infinite-dimensional Lie algebras. One of them is the paper Chevalley-Jordan decomposition for a class of locally finite Lie algebras by Ian Stewart. In general, there are several papers on infinite-dimensional Lie algebras, e.g., by Ivan Penkov, which use these decompositions.