How is the following property true? Let $I$ be the identity matrix and $A$, $B$ be appropriately sized real matrices. Then $$I \otimes \left(\left( I \otimes A\right) B \right) = \left( I \otimes I \otimes A \right)\left( I \otimes B\right).$$ I think I'm missing something basic about the Kronecker product.
The kronecker product is associative, so $(A\otimes B)\otimes C=A\otimes(B\otimes C)$ and use that $(AB)\otimes(CD)=(A\otimes C)(B\otimes D)$. Now, you can just expand.