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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 ...

In the paper [1], it is proven in Theorem 2 that any $n$-tensor $\mathcal{A}\in\mathbb{R}^{d_1\times...\times d_n}$ can be decomposed as $$ \mathcal{A}=\mathcal{S} \times_1 U_1 ...\times_n U_n $$ ...