Question on vector identity proof in order to derive maxwell stress tensor

Question

In the process of deriving the Maxwell stress tensor we have proven the following vector identity \begin{align} (\vec{B} \cdot \vec{\nabla})\vec{B} + \vec{B} \times (\vec{\nabla} \times \vec{B}) &= (B_j \partial_jB_i + \epsilon_{ijk}B_j\epsilon_{klm}\partial_lB_m)\vec{e}_i \\ &= (B_j\partial_jB_i + (\delta_{il}\delta_{jm} - \delta_{im}\delta_{jl})B_j\partial_lB_m)\vec{e}_i \\ &= (B_j\partial_jB_i + B_j\partial_iB_j - B_j\partial_jB_i)\vec{e}_i \\ &= B_j\partial_iB_j\vec{e}_i \\ &= \frac{1}{2}\vec{\nabla} (\vec{B}^2)\end{align}

I am wondering about what has happened in order to get the final "=". Can anyone explain it exactly?

Answer 1

Well, $$B_j\partial_iB_j\vec{e}_i=B_j\vec{\nabla}B_j=\frac{1}{2}\vec{\nabla}\left(B_jB_j\right)=\frac{1}{2}\vec{\nabla}\left(\vec{B}^2\right).$$