Conjugacy class in SO(4)

Question

Is it true that a matrix of the form $$ R = \begin{pmatrix} \cos \theta_1 & \sin \theta_1 & 0 & 0\\ -\sin \theta_1 & \cos \theta_1 & 0 & 0\\ 0 & 0 & \cos \theta_2 & \sin \theta_2\\ 0 & 0 & -\sin \theta_2 & \cos \theta_2 \end{pmatrix} $$ with $\theta_1+\theta_2=0$ is conjugate in SO(4) to a matrix of the form $$ R' = \begin{pmatrix} A & 0\\ 0 & 1 \end{pmatrix} $$ where $A$ is some 3x3 matrix and the 0's are column and row, namely that there exists $h \in SO(4)$ such that $R'=h^{-1} R h$ ?