Over an arbitrary ring $R$ with unit, is the matrix\begin{pmatrix} a & 0 & b & 0\\ 0 & 0 & 0 &0\\c & 0 & d &0\\ 0 &0&0&0 \end{pmatrix}
conjugate over $GL(R)$ to \begin{pmatrix} a & b & 0&0\\ c & d & 0&0\\0 & 0 &0 &0\\0&0&0&0 \end{pmatrix} ?
Over an arbitrary ring $R$ with unit, is the matrix\begin{pmatrix} a & 0 & b & 0\\ 0 & 0 & 0 &0\\c & 0 & d &0\\ 0 &0&0&0 \end{pmatrix}
conjugate over $GL(R)$ to \begin{pmatrix} a & b & 0&0\\ c & d & 0&0\\0 & 0 &0 &0\\0&0&0&0 \end{pmatrix} ?
View the matrices as linear maps to see that they are the same (one is the matrix of the other when we switch the second and third vectors). Thus take $P:=\pmatrix{\mathbf 1_R&0&0&0\\ 0&0&\mathbf 1_R&0\\ 0&\mathbf 1_R&0&0\\ 0&0&0&\mathbf 1_R},$ where $\mathbf 1_R$ is the unit of $R$.
You can get from one to the other by swapping the second and third rows and then the second and third columns. This corresponds to multiplying on the left by a certain row-switching matrix and then on the right by a certain column-switching matrix, as described in Wikipedia. What are they? Note that such matrices are their own inverses.