Show that $[L^2,[L^2,\vec{r}\,]\,] = 2\hbar^2\{L^2, \vec{r}\},$ where $\vec{r} = x\, {\hat x} + y\, {\hat y} + z\, {\hat z}.$

"Edit: $\{A,B\} = AB + BA$ is the anti-commutator."

I am able to solve this by brute force expanding everything, but I was wondering if there is a quicker, more elegant way of doing so. Also, if anyone has any physical intuition behind what this result means, it would be much appreciated.