Let $W_1,W_2,W_3 \in \mathbb{R}^{n,n}$ be such that
$$\eqalign{ \Big[\matrix{W_1^T&W_2^T&W_3^T}\Big]\cdot\left[\matrix{W_1\\W_2\\W_3}\right]\cdot = I }$$
where $I$ is the identity matrix. I'm a bit lost here but I expect then that the following holds for $A \in \mathbb{R}^{n,n}$
$$\eqalign{ \Big[\matrix{A^TW_1^T&A^TW_2^T&A^TW_3^T}\Big]\cdot\left[\matrix{W_1A\\W_2A\\W_3A}\right] = A }$$
Is that the case?
In general, $$ \pmatrix{A^TW_1^T&A^TW_2^T&A^TW_3^T}\pmatrix{W_1A\\ W_2A\\ W_3A} =A^T\pmatrix{W_1^T&W_2^T&W_3^T}\pmatrix{W_1\\ W_2\\ W_3}A =A^TA. $$ Whether $A^TA$ is equal to $A$ depends on the exact value of $A$ itself.