Show that $L/L'$ abelian

Question

Problem: Let $L$ be a Lie algebra, denote $[L L]=L'$. Show that $L/L'$ abelian.

My attempt: $[x,y] = (x+L')(y+L')-(y+L')(x+L') = ((x+y)+L') - (y+x+L') = ((x+y)+L') - ((x+y)+L') = 0$

Is that enough???