Vector calculus identity for $\nabla\times(\vec{b}\cdot\nabla)\vec{b}$

Question

I'm going through a paper on turbulence and in it the author uses the following $$ \nabla\times(\vec{b}\cdot\nabla)\vec{b}=(\vec{b}\cdot\nabla)(\nabla\times\vec{b})-\left((\nabla\times\vec{b})\cdot\nabla\right)\vec{b} $$ however I have tried to verify this with both vector analysis identities and using suffix notation and I can't seem to do so. I wondered if anyone could either show why it holds or correct it. Thanks in advance.

Answer 1

EDIT: It turns out the identity is incorrect. Consider the vector field $\vec{b}=\begin{pmatrix}xy\\0\\xy\end{pmatrix}$ (it doesn't work)

Answer 2

Extremely late comment, but this identity is true if the flow is incompressible, so $\nabla \cdot b =0$.