Evaluate or simplify two directional derivatives of the direction vector $(u\cdot \nabla )^2 u $

Question

Give a vector $\vec u$ that is a vector valued function of $\mathbb R^3$, $\vec u(x,y,z)$, what is the meaning of: $$ (\vec u\cdot \nabla )(\vec u\cdot \nabla )\vec u $$

Does this simplify to anything nice? So far I have that $$ (\vec u\cdot \nabla)\vec u = \frac{1}{2}\nabla (\vec u\cdot \vec u ) - \vec u \times ( \nabla \times \vec u) $$