Consider the group $SO(4)$. We know that each element in $SO(4)$ can be decomposed as a product of a left- and right-isoclinic rotation. Now, I have found two sets of three $SO(4)$ matrices that fulfill the $SU(2)$ algebra relations (as expected because $SO(4) \simeq SU(2) \otimes SU(2) / \mathbb{Z}_2$) and matrices of the different sets commute.
Now I would like to consider the subgroup of $SO(4)$ that is spanned by choosing multiplying "identical" left and right-isoclinic rotations, i.e. elements $G = L R$ where $L= a^0 1 + a^j M^j$ and $R = a^0 1 +a^j G^j$ where $M^j$ and $G^j$ are the respective elements of the two sets.
Specifically, I am interested in the center of the group described above. An explicit computation with Mathematica reveals that the center manifold is two-dimensional. I am curious if the subgroup described above has any geometrical interpretation/formal name, and if there are more rigorous results established.
Thanks!