How to write in closed form this nasty expression?

Question

I have something like $$ v_1 l_1+v_1 l_2+ v_2l_1+v_2l_2$$ and I am trying to write it in closed form as such, $$\sum_{j=1}\sum_{i=1}v_il_j$$ I know this is not right but I want something like that.

Also I would like to write in closed form somthing like $$v_{1}v_{2}+v_{1}v_{3}+v_{2}v_{3}+v_{2}v_{1}+v_{3}v_{1}+v_{3}v_{2}$$

Answer 1

Let $S = \{ (i,j) \in \mathbb{Z}^2: i \neq j, 1 \leq i \leq 3, 1 \leq j \leq 3\}$. Then

$$ v_1v_2 + v_1v_3 + v_2v_3 + v_2v_1 + v_3v_1 + v_3v_2 = \sum_{(i,j) \in S} v_iv_j $$