Closed subsets of root systems and regular subalgebras

We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Such a classification of an irreducible root system is closely related to the classification of the regular subalgebras, up to inner automorphism, of the corresponding simple Lie algebra. We implement our algorithm to classify the closed subsets of the irreducible root systems of ranks 3 through 7. We present a complete description of the classification for the closed subsets of the rank 3 irreducible root system. We employ this root system classification to classify all regular subalgebras of the rank 3 simple Lie algebras. We present only summary data for the classifications in higher ranks due to the large size of these classifications. Our algorithm is implemented in the language of the computer algebra system GAP.