How to find two sets of vectors which satisfy a set of matrix equations

Question

In my trial to solve a system of matrix equations, I wish to find two sets of non-zero vectors of $\mathbb{R}^3$ (which may be not unique) $\{ A_i \}$ and $\{ B_i \}$ where $i \in I$ (an index set, with a decomposition $I = I_1 \sqcup I_2$) satisfying $$ \tag{1} \ \sum_{ i \in I } A_i \otimes B_i - B_i \otimes A_i \ = \ \left( 0\text{-matrix} \right) $$ $$ \tag{2} \ \sum_{ i \in I_1 } A_i \otimes A_i - B_i \otimes B_i \ = \ \left( \text{Identity-matrix} \right) $$ $$ \tag{3} \ \sum_{ i \in I_2 } A_i \otimes A_i - B_i \otimes B_i \ = \ \frac{1}{2} \left( \text{Identity-matrix} \right). $$ where the matrices are $3 \times 3$-matrices, and $\left( A_i \otimes B_i \right)_{ lm }$ ( $l$-th rank and $m$-th column element of the matrix $A_i \otimes B_i$ ) is defined as $\left( A_i \right)_l \left( B_i \right)_m$ ( the multiplication of $l$-th element of vector $A_i$ with $m$-th element of vector $B_i$, where $1 \leq l, m \leq 3$.)