He is not claiming that the commutator is equivalent to the divergence, what he is saying is that [xl&#x27;,Hl] = 0 is equivalent to div H = 0, which is different. If you click on the annotation on the left you will see that [xl&#x27;,Hl] = 0 means that Hl is not a function of xl and so ∂_lH_l will be zero and consequently div H = Sum ∂_lH_l is also zero.