I was going over past PGRE exam questions, and came across this one.
The components for the angular momentum operator $\mathbf{L}=(L_x,L_y,L_z)$ satisfy the following commutation relations. \begin{align*} [L_x,L_y]&=i\hbar L_z\\ [L_y,L_z]&=i\hbar L_x\\ [L_z,L_x]&=i\hbar L_y \end{align*} What is the value of the commutator $[L_xL_y,L_z]$? (The answer was (D), $i\hbar(L_x^2-L_y^2)$)
This is easily solved by expanding the commutator, but I was very interested in one of the answers given on this website.
A quick way to do this: As the commutator is $[L_xL_y, L_z]$, Use the right hand rule to point first in the $+X,+Y$ (diagonal) direction, and then curl up to the $Z$ direction. Your thumb will be pointing in the $+X,-Y$ direction, so thus choice (D).
How did the reasoning in that person's answer work? Is there more (than tricky coincidence) behind what he said?