I am trying to understand the space of all orthogonal tensors, a question asked [here][1] 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?

  [1]: https://mathoverflow.net/questions/361610/characterization-of-all-orthogonal-tensors