Let $W = [W_1,W_2,W_3]^\top$ where $W_1,W_2,W_3 \in \mathbb{R}^{n,n}$. My question is if $(W \otimes W)^\top(W \otimes W)$ is the same as
$$ \left[ \begin{array}{c c} W_1^\top \otimes W_1^\top& W_1^\top \otimes W_2^\top& W_1^\top \otimes W_3^\top& W_2^\top \otimes W_1^\top& W_2^\top \otimes W_2^\top& W_2^\top \otimes W_3^\top& W_3^\top \otimes W_1^\top& W_3^\top \otimes W_2^\top& W_3^\top \otimes W_3^\top \end{array} \right] \left[ \begin{array}{c c} W_1 \otimes W_1 \\ W_1 \otimes W_2 \\ W_1 \otimes W_3 \\ W_2 \otimes W_1 \\ W_2 \otimes W_2 \\ W_2 \otimes W_3 \\ W_3 \otimes W_1 \\ W_3 \otimes W_2 \\ W_3 \otimes W_3 \\ \end{array} \right] $$
My understanding is that this equality follows from the basic rules of the Kronecker product.
From the definition of the matrix $W$ note that $$\eqalign{ W^TW &= \Big[\matrix{W_1&W_2&W_3}\Big]\cdot\left[\matrix{W_1^T\\W_2^T\\W_3^T}\right] \\ &= \Big(W_1W_1^T + W_2W_2^T + W_3W_3^T\Big) \\&= \sum_{i=1}^3W_iW_i^T \\ }$$ Expanding the Kronecker expression in question yields $$\eqalign{ \big(W\otimes W\big)^T\big(W\otimes W\big) &= \left(W^T\otimes W^T\right)\big(W\otimes W\big) \\&= W^TW\otimes W^TW \\ &= \sum_{i=1}^3\sum_{j=1}^3W_iW_i^T\otimes W_jW_j^T \\ }$$ and expanding the sums leads to $9$ terms like $\;W_2W_2^T\otimes W_3W_3^T$