Let $\mathfrak g$ be a Lie algebra and let $S$ be a subset of $\mathfrak g$. How to show that the Lie subalgebra generated by $S$ consists of all linear combinations of the elements $[s_m, s_{m−1}, ..., s_1]$, where $m≥ 1$ and $s_i\in S$?
Where we define $[x_1]=x_1$ and $[x_n, x_{n−1}, ..., x_1] = [x_n, [x_{n−1}, ..., x_1]]$, for $ n ≥ 1$ and $x_1,...,x_n\in \mathfrak g$.