I am new to subscript notation, and I am having a problem with the proof practice below, $$\frac{1}{2}\vec{\nabla}(\vec{f}\cdot\vec{f}) = \vec{f} \times \vec{\nabla} \times \vec{f} + (\vec{f} \cdot \vec{\nabla})\vec{f}$$
I started from the first term RHS, and write it as below, $$\begin{aligned}\vec{f}\times\vec{\nabla}\times\vec{f} &= \epsilon_{ijk}f_i\nabla_{j}\epsilon_{klm}f_m \\ &=\epsilon_{ijk}\epsilon_{klm}f_i\nabla_jf_m \\ &=(\delta_{im}\delta_{jl}-\delta_{il}\delta_{jm})f_i\nabla_jf_m \\ &=\delta_{im}\delta_{jl}f_i\nabla_jf_m - \delta_{il}\delta_{jm}f_i\nabla_jf_m \\ &=f_m\nabla_jf_m - f_l\nabla_jf_j \\ &=f\cdot f\nabla -f(\nabla\cdot f) \end{aligned}$$
Is this correct? How to continue the proof?
Thank you very much!