I'm having some trouble using index notation to prove the identity
$$\frac{1}{2}\mathbf{\nabla (u \cdot u) = u \times (\nabla \times u ) + (u \cdot \nabla)u}$$
The closest I can get is by expanding the first term on the RHS, which gives
$$\mathbf{u \times (\nabla \times u)} = 2u_j \partial x_i u_j - u_j\partial x_i u_i - u_i \partial x_j u_j$$
but I don't see what to do from here (if what I've done so far is correct).
Any help will be appreciated!
EDIT
The comments so far are all a bit dubious about my expression for the first term on the RHS, here's my work:
$$\mathbf{u \times (\nabla \times u)} = \epsilon_{ijk}u_j\epsilon_{klm}\partial x_lu_m$$ now I move the second Levi-Civata symbol to the left and use the identity GFR posted to get
$$\epsilon_{ijk}u_j\epsilon_{klm}\partial x_lu_m = (\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})u_j\partial x_lu_m$$ expanding this gives
$$\mathbf{u \times (\nabla \times u)} = u_j\partial x_i u_j - u_j\partial x_j u_i$$ This next step i'm not sure about, I move the derivatives to the left of each term
$$u_j\partial x_i u_j - u_j\partial x_j u_i = \partial x_iu_ju_j - \partial x_j u_ju_i$$
then the product rule gives my original equation for $\mathbf{u \times (\nabla \times u)}$