Is it true that $ AB^TBA^T = BAA^TB^T $ From this answer on Stats Stackexchange https://stats.stackexchange.com/a/353715/274404, I don't agree with the last part.
\begin{align}AB^TBA^T
&= \left(\left(AB^TBA^T\right)^T\right)^T \\
&= \left(BA^TAB^T\right)^T \\
&= B\left(BA^TA\right)^T \\
&= BAA^TB^T
\end{align}
If I consider that B is a square matrix and A is the identity matrix, it would imply that $ B^T B= B B^T$. Am I missing something here? 

Your suspicion is correct: the author of the linked answer was wrong. In fact, in that answer, if $AZ$ is a valid matrix-vector product, then $ZA^T$ doesn't make sense unless $Z$ is $1\times1$.
