Lie algebra vector subspace: Does $[n_1,[n_2,Y]]=[n_1,A]=B$

Question

If $Y$ is a vector subspace of the Lie algebra $\mathfrak{g}$ and $n_1,n_2\in N_\mathfrak{g}(Y)$ does the following hold?

$$[n_1,[n_2,Y]]=[n_1,A]=B$$ where $B\subseteq A\subseteq Y$

Answer 1

The idea that you are looking for is that $[n_1,[n_2,Y]]\subseteq [n_1,Y] \subseteq Y$ Your placeholders hold, but in the abstract case they can't be found of course.