I am trying to understand the space of all orthogonal tensors, a question asked here before but with no real solution yet found. The solutions for order 2 tensors are clear so thus the simplest case is a 2x2x2 tensor (with complex values). This means if the 2x2x2 tensor is denoted by $a_{i,j,k}$, then the following three equations must hold:

$ \sum_{i=1}^{2} \sum_{j=1}^{2} a_{i,j,1} \overline{a_{i,j,2}} = 0 $

$\sum_{i=1}^{2} \sum_{k=1}^{2} a_{i,1,k} \overline{a_{i,2,k}} = 0 $

$\sum_{j=1}^{2} \sum_{k=1}^{2} a_{1,j,k} \overline{a_{2,j,k}} = 0 $

How can one characterize the space of solutions to these 3 equations?

I am trying to understand the space of all orthogonal tensors, a question asked here before but with no real solution yet found. The solutions for order-$2$ tensors are clear so thus the simplest case is a $2\times2\times2$ tensor (with complex values). This means if the $2\times2\times2$ tensor is denoted by $a_{i,j,k}$, then the following three equations must hold:

$$ \sum_{i=1}^2 \sum_{j=1}^2 a_{i,j,1} \overline{a_{i,j,2}} = 0 $$

$$\sum_{i=1}^2 \sum_{k=1}^2 a_{i,1,k} \overline{a_{i,2,k}} = 0 $$

$$\sum_{j=1}^2 \sum_{k=1}^2 a_{1,j,k} \overline{a_{2,j,k}} = 0 $$

How can one characterize the space of solutions to these three equations?