Let $T$ be a $(p, q)$-tensor field, and let $X$ and $Y$ be vector fields. Show that the Lie derivative satisfies $\left[\mathcal{L}_{X}, \mathcal{L}_{Y}\right] T=\mathcal{L}_{[X, Y]} T$. It is enough to consider the cases

(i) $(p, q)=(1,0)$, i.e., $T$ is a vector field, and

(ii) $(p, q)=(0,1)$, i.e., $T$ is a cotangent vector field.

The general case follows by applying the Leibniz rule.