Is there any intuition why the following matrix is positive semidefinite?

If $A$ is the $4\times 4$ submatrix in the upper left corner and $J$ is the negative permutation matrix $$ J=\begin{bmatrix} &&&-1\\ &&-1&\\ &-1&&\\ -1&&& \end{bmatrix} $$ then the original matrix is $$ \begin{bmatrix} A & AJ\\ JA & JAJ \end{bmatrix}= \begin{bmatrix} I\\J\end{bmatrix}A \begin{bmatrix} I &J\end{bmatrix} $$ which is positive semidefinite if $A$ is. The positive semidefiniteness of $A$ follows e.g. from $$ A=2(I+J)+ee^T $$ where $e$ is the vector of all ones.