The question is:
Using Cartesian coordinates, show that
(u ∇)u = 1/2 ∇(u u) − u ( ∇ u) ,
and hence that
∇ ((u ∇)u) = ( ∇ u)( ∇ u) + (u ∇)( ∇ u) − (( ∇ u) ∇)u .
I can do the first part using suffix notation fine, but I can't see how it really helps with the 2nd part. I just seem to be going around in circles. On the LHS I get:

\epsilon_{ijk} \frac{\partial}{\partial x_{j}} (u_{l}\frac{\partial}{\partial x_{l}}) u_{k}

but I can get this without using the 1st identity. Working backwards from the RHS I'm getting a bit muddled in all the subscripts. Any ideas?