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