$$ \begin{align} [xL_x, L_z] &= [x, L_z]L_x + x[L_x, L_z] \\ &= -i{\hbar}yL_x - i{\hbar}xL_y \end{align} $$
$$ \begin{gather} [x, L_z] = -i{\hbar}y \\ [\alpha, L_{\beta}] = i{\hbar}\epsilon_{\alpha\beta\gamma}\gamma \\ [L_x, L_z] = -i{\hbar}L_y \\ \end{gather} $$
$$ \begin{align} [xL_x, L_z] &= [x, L_z]L_x + x[L_x, L_z] \\ &= -i{\hbar}yL_x - i{\hbar}xL_y \end{align} $$
$$ \begin{gather} [x, L_z] = -i{\hbar}y \\ [\alpha, L_{\beta}] = i{\hbar}\epsilon_{\alpha\beta\gamma}\gamma \\ [L_x, L_z] = -i{\hbar}L_y \\ \end{gather} $$
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