We prove that there is exactly one homothety class of invariant Einstein metrics in each space $SU(2) \times SU(2) / SO(2)_r (r\in Q, \, |r|\neq 1)$ defined below.
@article{116981, author = {Eugene D. Rodionov}, title = {Einstein metrics on a class of five-dimensional homogeneous spaces}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {32}, year = {1991}, pages = {389-393}, zbl = {0747.53037}, mrnumber = {1137801}, language = {en}, url = {http://dml.mathdoc.fr/item/116981} }
Rodionov, Eugene D. Einstein metrics on a class of five-dimensional homogeneous spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 389-393. http://gdmltest.u-ga.fr/item/116981/
Einstein manifolds, Springer Verlag, Berlin, 1987. | MR 0867684 | Zbl 1147.53001
Homogeneous Einstein spaces of dimension $4$, J. of Diff. Geom. 3 (1969), 309-349. (1969) | MR 0261487
Classification of five-dimensional naturally reductive spaces, Math. Proc. Camb. Phil. Soc. 97 (1985), 445-463. (1985) | MR 0778679 | Zbl 0555.53024