This is an exercise in Humphrey's 'Introduction to Lie Algebras and Representation Theory' (chapter 1.2 number4).
Here is what I've done.
Since $[LL]$ has dimension 1, let {$x$} be a basis for $[LL]$. Extend it to a basis {$x, y, z$} for L.Then $[xy]=ax, [yz]=bx$ and $[xz]=cx$. (Not all $a, b, c$ are zeros.) Since [$LL$] lies in $Z(L)$, $0=[[xy]z]=a[xz]$ and $0=[[xz]y]=c[xy]$. Then we haver four cases where each factor is zero or not zero.
If $[xz]=[xy]=0$, then $[yz]=bx$ should not be zero.
If $a=c=0$, then again, $[xz]=[xy]=0$ and $[yz]=bx$ should not be zero.
Now, here is where I stucked. If '$a=0$ and $[xz]\neq 0$' or '$a \neq 0$ and $[xz]=0$', then how can I get the result "$[yz]=bx$ is not zero."? Or is there better way to solve this problem?
Thanks in advance.
