I have a question about a property of Jordan Chevalley decomposion components of any element of a semisimple Lie subalgebra $g \subset gl(V)$.
If we have for any $x \in g$ a J-C decomposion $ x= x_s +x_n$ why there are holding the inclusions $ad(x_s)(g) \subseteq g$ and $ ad(x_n)(g) \subseteq g$, where for $w,v \in g$ the $ad$ function is defined as $ad(w)(v) := [w,v]$?
Because $x_s,x_n\in g$. You will find a proof of this assertion in Bourbaki's textbook on Lie groups and Lie algebras (chap. VII, §5), for instance.