Let $L$ be a Lie algebra over the field $F$ with the property that $\forall x,y,z\in L ; [[x,y],z]=0$ . How to show that $L^3=0?$ Here $L^3=[L,L^2],L^2=[L,L^1],L^1=[L,L].$
There is no condition on the dimension of the Lie algebra over the field $F,$ $Char(F)\neq 3.$
Help me to solve this .