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Given a natural $n>2$, I want to show that there exists a lie algebra $g$ which is solvable of derived length 2, but nilpotent of degree $n$.

I have seen a parallel idea in groups, but i can't see how i can implement it for Lie-algebras.

Thanks!

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The so called standard graded filiform nilpotent Lie algebra $\mathfrak{f}_{n+1}$ of dimension $n+1$ has nilpotency class $n$, and derived length $2$. The non-trivial brackets are $[e_1,e_i]= e_{i+1}$ for $i=2,\ldots ,n$. We have $[\mathfrak{f}_{n+1}, \mathfrak{f}_{n+1}]=\langle e_3,\ldots ,e_{n+1}\rangle$ and $[[\mathfrak{f}_{n+1}, \mathfrak{f}_{n+1}], [\mathfrak{f}_{n+1}, \mathfrak{f}_{n+1}]]=0$.

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