\left[L_{x},L_{y}\right]=\left[yp_{z}-zp_{y},zp_{x}-xp_{z}\right]
=\left[yp_{z},zp_{x}\right]-\left[zp_{y},zp_{x}\right]-\left[yp_{z},xp_{z}\right]+\left[zp_{y},xp_{z}\right]
How next?
My book is not of much help
I Tried
\left[A,BC\right]=\left[A,B\right]C+B\left[A,C\right]
But...