If x is in the derived algebra, show that Tr(ad a)=0

Question

Let $L$ be a Lie algebra, and let $a\in [L,L]$. How to prove that $trace(ad_a)=0$?

Answer 1

Hint: If $x=[a,b]$, Jacobi implies that $ad_x=ad_a(ad_b)-ad_b(ad_a)$. Since $trace(fg)=trace(gf)$ the result follows.