Let $L$ be a Lie-Algebra over $k$, $gl(k)$ as usual. Let $ad_x:L \rightarrow L$, given by $y \mapsto [x,y]$. Show that $ad_x$ is Derivation, i.e. $ad_x([y,z])=[ad_x(y),z]+[y,ad_x(z)]$.
I think the way to go here is using the Jacobi identity:
$$[x,[y,z]]+[z,[x,y]+[y,[z,x]]=0 \leftrightarrow ad_x([y,z])=-[z,[x,y]-[y,[z,x]]$$
Now using the Antisymmetric, we get $-[z,[x,y]]=[[x,y],z]=[ad_x(y),z]$ (what to do if $char(k)=2$?). So I have to show that $-[y,[z,x]]=[y,ad_x(z)]$. However, I get $-[y,[z,x]]=[-[x,z],y]$ and I do not know how to go on.