Consider the operator, $\mathbf{L} = -i\hbar\mathbf{r}\times \nabla$ where $i = \sqrt -1$. We need to prove that $\mathbf{L}\times\mathbf{L}f = i\hbar\mathbf{L}f$ , where $f$ is an arbitrary test function.
Applying basic vector identities related to $\nabla$, I have reached the following result: $\mathbf{L}\times\mathbf{L}f = 2i\mathbf{L}f + \mathbf{r}\times(\mathbf{r}\cdot\nabla)(\nabla f) $ (consider $\hbar = 1$)
I am not able to reduce the second term, any hints are appreciated.
(Preferably without the use of Levi-Civita and brute force expansion)
I had also posted this on Physics SE, but the people there weren't exactly helpful.