Commutators of dot and cross products

Question

I apologize if this question is too basic, but I am wondering if identities for commutators such as $[AB,C]=A[B,C]+[A,C]B$ also hold for dot and cross products within the commutator (i.e., $[\vec{A}\times \vec{B},C]=\vec{A}\times[\vec{B},C]+[\vec{A},C]\times \vec{B}$, and same for dot product). I've tried writing it through in component form and it seems like it checks out okay, but I would really like to double check that I'm not making a stupid mistake and thinking wishfully, as I frequently lose myself in long chains of algebra.

Answer 1

It looks like the identities you want to prove are$$\left[\sum_iA_iB_i,\,C\right]=\sum_iA_i[B_i,\,C]+\sum_i[A_i,\,C]B_i,\,\\\left[\sum_{jk}\epsilon_{ijk}A_jB_k,\,C\right]=\sum_{jk}A_j[B_k,\,C]+\sum_{jk}[A_j,\,C]B_k.$$These just follow from the usual identity, with one of the operators $\sum_i,\,\sum_{jk}\epsilon_{ijk}$ applied to it. For example,$$\left[\sum_iA_iB_i,\,C\right]=\sum_i\left[A_iB_i,\,C\right]=\sum_i\left(A_i\left[B_i,\,C\right]+\left[A_i,\,C\right]B_i\right).$$