In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$ related to a basis $\Delta$ for the root system $\Phi$ associated with $W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We show that the set of $\theta$-twisted involutions in $W$, $\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one to one correspondence with the set of regular involutions $\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are characterized by sequences in $\Sigma$ which induce an ordering called the Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.