I have been given the problem of showing that nilpotent Lie algebras are also solvable. While the proof as a whole is not difficult, I am struggling to understand the induction involved in the solution I have been given. I get they have used the induction assumption but I'm struggling to understand how they have concluded that $\mathfrak{g}^{(n)}\subseteq \mathfrak{g}^{n}$. Could anyone shed some light on this? Thanks!
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Because of $\mathfrak{g}^{(i)}\subseteq \mathfrak{g}^i$ for all $i\ge 1$, every nilpotent Lie algebra is solvable.
For $i=1$ this is clear since $\mathfrak{g}^{(1)}= \mathfrak{g}^1=\mathfrak{g}$. Now do the induction step $i\mapsto i+1$. \begin{align*} \mathfrak{g}^{(i+1)} & = [\mathfrak{g}^{(i)},\mathfrak{g}^{(i)}]\subseteq [\mathfrak{g}^i,\mathfrak{g}^i]\subseteq [\mathfrak{g},\mathfrak{g}^i]=\mathfrak{g}^{i+1}. \end{align*}
Dietrich Burde
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