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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.


marked as duplicate by Robin Ekman, AccidentalFourierTransform, John Rennie quantum-mechanics Apr 26 '16 at 6:12

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