Let $L$ be a finite dimensional $K$-Lie algebra and let $X$ be a subset of $L$. Ley $Y= \langle X \rangle _{K} $ the linear span of $X$. Also let $v_1,…,v_n $ be a basis of $Y$ and let $Z=\{[v_i , v_j ] : 1\leq i < n \leq n \} $. I need to show that $Y+\langle Z \rangle _K $ is contained in the subalgebra generated by $X$.
The proof I’m reading says that if $x,y \in Y$, then $ [x,y ] \in \langle X \rangle _{K-\text{alg}} $ but I can’t see why that has to be the case.