Let's say I have a subset of the generators for a real, semisimple Lie algebra. To find the full set of generators, I can take the Lie bracket of every element in my subset until it becomes closed. For any Lie algebra, there should be a minimal subset (maybe not unique) with the property that I recover the full Lie algebra by recursively applying the Lie bracket.
Does this subset have a name?
Is the form or even the cardinality of this subset known for any of the real semisimple Lie algebras?