In the definition of an isomorphism of root systems, Humphreys emphasizes that it is not assumed that the map is an isometry, it should just preserve the Cartan integers. I don't understand why we want this greater generality. If we have an isomorphism $L \to L'$ of Lie algebras, pick a maximal toral subalgebra $H \subset L$ and take its image $H'$ in $L'$, then the induced isomorphism of associated root systems IS an isometry. Also, in the discussions of lifting isomorphisms from root systems to Lie algebras in Section 14, there's this strange "it does no harm to assume that the isomorphism $\Phi \to \Phi'$ comes from an isometry". So, essentially, just the lifting of isometries is discussed. So, why do we consider more general isomorphisms of root systems? (I know that scaling of roots would be a "natural" isomorphism of root systems, but this does not lift to Lie algebra, does it?)
Isomorphism between root systems: why not assume isometry
lie-algebrasroot-systems
Best Answer
If we separate the root systems from the Lie algebras and view them as abstract objects, then we don't want to regard root systems obtained by scaling the roots as different. If we did that, then there would be uncountably many isomorphism classes of root systems. The goal is to get the isomorphism classes to be as simple as possible, in particular to obtain the classification by Dynkin diagrams using only the elementary axioms for root systems as subsets of a vector space as the starting point. We are taking something that arises in a non-elementary context (the root system of a Lie algebra) and abstracting out the properties that are necessary to classify it.