Write a complex finite-dimensional Lie algebra as $L=S\ltimes R$ (Levi decomposition). Then the subalgebra $[S,R]$ (generated by brackets $[s,r]$, $s\in S$, $r\in R$) is an ideal, by a simple verification based on the Jacobi and the equality $[S,R]=[S,[S,R]]$.

Is there a classical reference for this probably well-known fact ($[S,R]$ is an ideal)?

Remark: I indeed had in mind the proof given by Robin Goodfellow and I'm pretty sure I once saw it written. Moreover $[S,R]$ can be defined with no reference to any $S$: this is the intersection $R\cap\bigcap_n L^{(n)}$, where $\bigcap_n L^{(n)}$ is the intersection of the derived series (and thus the largest perfect subalgebra in $L$, which is also the ideal generated by $S$).

Lie Algebras(now in Dover reprint): in your notation, $[L,L] \cap R = [L,R]$, which is therefore an ideal of $L$. It's a quick argument not using $[S,S]=S$ or the Jacobi identity. The fact that $[S,R]$ is an ideal takes a little more argument but is also an elementary consequence of Levi's theorem. (Note that Mal'cev's conjugacy theorem for Levi subalgebras isn't used.) $\endgroup$ – Jim Humphreys Nov 28 '16 at 22:56