Prove, by induction on $k$, that the following result holds for $\omega$ a $k$- form on $\mathbb R^n$
$$(L_\mathbb XL_\mathbb Y-L_\mathbb YL_\mathbb X)\omega=L_{[\mathbb X,\mathbb Y]}\omega .$$
Let $\omega=fd\alpha$, where $f$ is a smooth function and $\alpha$ is a $(k-1)$- form.
Assume that the result is true for $(k-1)$- forms, i.e
$$(L_\mathbb XL_\mathbb Y-L_\mathbb YL_\mathbb X)\alpha=L_{[\mathbb X,\mathbb Y]}\alpha .$$
We also know that
$$(L_\mathbb XL_\mathbb Y-L_\mathbb YL_\mathbb X)f=L_{[\mathbb X,\mathbb Y]}f.$$
I am struggling to show that
$$(L_\mathbb XL_\mathbb Y-L_\mathbb YL_\mathbb X)fd\alpha=L_{[\mathbb X,\mathbb Y]}fd\alpha .$$
I have that at the moment that
$$L_{[\mathbb X,\mathbb Y]}fd\alpha=(L_{[\mathbb X,\mathbb Y]}f)d\alpha+fL_{[\mathbb X,\mathbb Y]}d\alpha=((L_\mathbb XL_\mathbb Y-L_\mathbb YL_\mathbb X)f)d\alpha+f(L_\mathbb XL_\mathbb Y-L_\mathbb YL_\mathbb X)d\alpha$$.