I was hoping there is a textbook focused on Lie groups that has particular focus on the Riemannian geometry side of Lie groups: Left-invariant, bi-invariant metrics, relationship between the Lie and manifold exponential maps, left-invariant affine connections, the Cartan connection on Lie groups, etc. To my knowledge, although there are some great textbooks on Lie groups out there, I haven't found any that really focus on the Riemannian geometry properties of Lie groups. They usually introduce Lie groups, focus on their algebraic properties, talk about the Lie exponential, Lie algebra, classification. At least in my experience, nearly every textbook on Lie groups focus entirely on their algebraic properties.
The best I have found is the chapter on Lie groups in Lee's Introduction to Riemannian Manifolds. Of course, this is only a chapter. I'd appreciate a whole textbook about this.