- For regular subalgebras $h$ of some group's Lie algebra $g$, $$ h \subset h $$ the root system of the subalgebra is a subset of the root system of the original's group algebra.
- Subalgebras whose root system is not a subset of the root system of the original algebra are called special subalgebras
Nevertheless, there must be some map from the root system of the subalgebra to the root system of the original algebra because that's how embedding a subalgebra is defined. My problem is finding this map.
In other words: Given a set of roots for the original algebra $g$ and a subset of this root system related in some way to the root system of the special subalgebra. This subset is, by definition of a special subalgebra, not directly the root system of the subalgebra, but there must be some map to the correct root system of the special subalgebra. How can I find the corresponding map?