One of the cross product properties I am referring too is the following $$A \times (B \times C)= B(A \cdot C) - C(A \cdot B)$$
Now consider this identity:
$$V( \nabla \cdot V)= \nabla(V^2/2) - V \times \omega$$
Where $\omega=(\nabla \times V)$
$V$ is vector and $\nabla$ is the gradient. I don't understand this identity since the left hand side is not divided by $2$.
I have written as Letting $A=\nabla$ $B=\nabla$ and $C=V$ I can rewrite as $V \times ( \nabla \times V)= \nabla(V \cdot V) - V(\nabla \cdot V)$
Update: I just found the property being used and I am not even sure how to prove it.
$\nabla( V \cdot V)= V( \nabla \cdot V) + (V \cdot \nabla)V + V\times (\nabla \times V) +V \times (\nabla \times V) $
This explains where the one half is coming from but how to prove this is a mystery