Eigenvalue decomposition of Cross Gramian matrix

For a symmetric state space system $G(s)=\left\{A,B,C,D\right\}$, the cross Gramain matrix $R$ is the solution of $$AR+RA+BC=0$$

Using eigenvalue decomposition, problem is to obtain a matrix U which diagonalizes the cross gramian matrix $R$, resulting a diagonal matrix $S$ such that $$S=U^{-1}RU$$

Note: For state space symmetric system, $A=A^T, C=B^T$ where $T$ is the transpose of a matrix.