I'm working on question 4.8 on page 36 of Erdmann and Wildon's book called Introduction to Lie Algebras. The question is as follows:
Let $L$ be a Lie algebra over a field $F$, such that $[a,b],b]=0$ for all $a,b \in L$.
(i) If $Char(F) \neq 3$, then show that $L^3=0$;
(ii) if $F$ has characteristic 3 then show that $L^4=0$. Here $L^k=[L L^{k-1}]$ and $L^1=[L L]$.
Now to the key is to look at $[[a,b+c],b+c]$ and deduce that $[[a,c],b]=-[[a,b],c]$, and then use Jacobi to show that $[[a,b],c]=0$ for all $a,b,c \in L$. But doesn't this mean that $L^2=0$?. I'm just a little worried that I seem to have proven something stronger than the question asks, which makes me uneasy. Similarly for part (ii) I think the proof shows $L^3=0$.
So I was wondering if there is something I am overlooking.
Thank you
This CW answer intends to remove this question from the Unanswered queue.
As already noted in the comments, you are correct. It seems like the authors themselves got confused with the definitions.