For each of the two reference books the constant equations are as follows: $$ \boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}+\frac{1}{2}\nabla \boldsymbol{E}^2 $$

$$ \boldsymbol{E}\times \left( \nabla \times \boldsymbol{E} \right) =-\left( \boldsymbol{E}\cdot \nabla \right) \boldsymbol{E}+\left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla $$ We can reason about the terms that correspond to the same in both equations. $$ \left( \boldsymbol{E}\cdot \boldsymbol{E} \right) \nabla =\frac{1}{2}\nabla \boldsymbol{E}^2 $$ What are some ways to reason about the above equation to arrive at it? (I would like to have access to a variety of reasoning options)