Einstein warped product spaces on Lie groups

We consider a compact Lie group with bi-invariant metric, coming from the Killing form. In this paper, we study Einstein warped product space, $M = M_1 \times_{f_1} M_2$ for cases, $(i)$ $M_1$ is $(ii)$ $M_2$ and $(iii)$ both are groups. Moreover, obtain conditions an of groups to become simple manifold. Then, characterize warping function generalized Robertson-Walker spacetime, $(M I G_2, - dt^2 + f_1^2 g_2)$ whose fiber $G_2$, being semi-simple $\dim G_2>2$, having