Regarding lower central series of a Lie algebra L

Question

Let L be a Lie algebra and let $(L^i)$ be its lower central series defined by $L^1=L$, $L^i=[LL^{i-1}]$. Then it can be proved by induction that $[L^iL^j] \subseteq L^{i+j}$. Please help me to find an example in which the above containment is proper.

Answer 1

Take the standard graded filiform nilpotent Lie algebra $L_n$, with basis $(e_1,\ldots ,e_n)$ and Lie brackets $[e_1,e_i]=e_{i+1}$ for $2\le i\le n-1$. Then $L^1=L$, $L^i=\langle e_{i+1},\ldots ,e_n\rangle$ for $i\ge 2$. Now $[L^2,L^2]=0$ but $L^{2+2}=L^4=\langle e_5,\ldots ,e_n\rangle$ for $n\ge 5$.