Let $L$ be a finite dimensional semisimple Lie algebra over a field of charcteristic $0$ and algebraically closed. I have learned that the map $$ ad: L \rightarrow \ Der(L) $$ is an isompsphism, where $adx(y) = [x,y]$ and $Der(L)$ denotes all the derivations on $L$. Furthermore, I have learned that if $x \in L$ is semisimple then $adx:L \rightarrow L$ is semisimple. I was wondering from these information does it actually follow that $x_s \in L$? (where $x_s$ is the unique semisimple part of $x$)
Clarification: Given $x \in L$ we can always write it as $x = x_s + x_n$ where $x_s$ is the semisimple part and $x_n$ is the nilpotent part and I know that they both lie in $gl(V)$ but do they lie in $L$? (and I am viewing $L$ as a subalgebra of $gl(V)$ for some finite dimensional vector space $V$)