http://www.unl.edu.ar/ceneha/uploads/Cartesian_tensors_Index_notation_&_summation_convention.pdf avers:
$1.$ $(a×b).(c×d) = \epsilon_{i jk}a_jb_k \quad e_{ilm}c_ld_m$
$2. \nabla × \color{tomato}{F×G}_i = e_{i jk} \partial_ j \color{tomato}{e_{klm}F_lG_m}$. (This is Stewart P1107 Ch 16 Review 20)
But by definition, $a \times b = \epsilon_{i jk}a_jb_k\color{forestgreen}{\mathbf{e_i}}$? Why is $\color{forestgreen}{\mathbf{e_i}}$ missing?
$2.$ The placement of the $i$ is ambiguous. Does it denotes the ith component of the entire expression, so it should've been written $[ \; \nabla × \color{tomato}{F×G} \;]_i$ ? Again, where's $\color{forestgreen}{\mathbf{e_i}}$?
$3.$ I see the same problem in http://www.stanford.edu/~vkl/research/notes/index_not.pdf:
$\mathbf{u} \times \mathbf{v} = \epsilon_{\color{green}{i} \, jk}u_jv_k$
$\text{curl} \,\mathbf{u} = \nabla \times \mathbf{u} = \epsilon_{\color{green}{i} \quad jk}\partial_jv_k$
Shouldn’t these expressions contain $\color{forestgreen}{\mathbf{e_i}}$?
On cases 1. and 2. A more understandable notation would be (we use Einstein convention on repeated indices) $$(a\times b).(c\times d) :=(\text{this is the canonical scalar product})= (a\times b)_i(c\times d)_i=\epsilon_{i jk}\epsilon_{ilm}a_jb_k c_ld_m,$$
and
$$ \left(\nabla \times ( {F\times G})\right)_i = \epsilon_{i jk}\epsilon_{klm} \partial_ j(F_lG_m),$$
for all $i=1,2,3$. We are then considering the $i$-th component of the vector $\nabla \times ( {F\times G})$.
An important remark: the cross product in not associative; so the bracket in $\nabla \times ( {F\times G})$ becomes important. As it is missing, this is a mistake.
On case 3. The notation is sloppy; on the l.h.s. of both cases we have a vector, while on the r.h.s. we have the $i$-th component of it. It has to be corrected.