Let $\{e,h,f\}$ be the standard basis of the Lie algebra $\mathfrak{sl}(2,k)$. Prove that $(\mbox{ad }e)^3=0$
http://en.wikipedia.org/wiki/Special_linear_Lie_algebra
First I computed $(\mbox{ad }e)(y)$. Let $y=ah+be+cf$, then $(\mbox{ad }e)(y)=[e,y]=ch-ae$. However, I don't really know what $(\mbox{ad }e)^3=0$ means in notation as we only defined $\mbox{ad}$ is not to the power of three.
So anyone got any ideas on what it means and how you actually prove it?