Are there any relation between Einstein structure on a Lie group and chirality of its Lie algebra?

Question

An Einstein structure is a  Riemannian metric on a manifold M such that $Ric=k g$   for  a  constant  $k$. $Ric$ is the Ricci tensor and $g$ is the metric tensor

https://link.springer.com/book/10.1007/978-3-540-74311-8

Are  there  any relations between existence of  an  Einstein structure on a Lie  group  and  chirality of its  Lie  algebra? Under  what  conditions  a Lie  group whose  Lie  algebra is  a  chiral algebra admit an Einstein structure?

Comments

What is an Einstein structure? Please give a reference.

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@ArnoldNeumaier   Thank you for your comment. An Einstein structure is a  Riemannian metric on a manifold M such that Ric=k g   for  a  constant  k. Ric is the Ricci tensor and g is the metric tensor

https://link.springer.com/book/10.1007/978-3-540-74311-8