External Algebra – A Vector Computation Formula

I have unsuccessfully asked this question about mathematics. Let $ A, B $ to be vector fields in $ mathbb R ^ 3 $. We have
$$
text {curl} bigl ((A cdot nabla) B bigr) = (A cdot nabla) text {curl} B
– (( text {curl} A) cdot nabla) B + R (A, B).
$$

I know it $ R (A, A) = $ 0 and I guess $ R $ should look like
$$
d a_j wedge
d b_j
= text {curl} A times text {curl} B + text {div} A text {curl} B
– text {div} B text {curl} A. $$

Is there a simple closed form for $ R (A, B) $?