Rootsystems of simple Lie algebras

Abstract

Rootsystems of nonclassical simple Lie algebras L = [summation operator]a[epsilon]R La such that a([e,f]) [not equal to] 0 for some e[epsilon]L'a, f[epsilon]L'-a for each a[epsilon]R - {0} either contain T2-sections or are irreducible Witt rootsystems. The irreducible Witt rootsystems of prime ranks 1, 2, 3 are W, W2, S2, W3, W[plus sign in circle](W [logical or] W), W[plus sign in circle]S2, S3, S3[plus sign in circle](W [logical or] W), S3(S2). Witt rootsystems having no sections S2, W[plus sign in circle](W [logical or] W) are classified as those rootsystems whose irreducible components are finite vector space subgroups. Since the latter are rootsystems of generalized Albert-Zassenhaus Lie algebras, it follows that the rootsystems of nonclassical simple Lie algebras L = [summation operator]a[epsilon]R La such that ([e,f]) [not equal to] 0 for some e[epsilon]L'a, f[epsilon]L'-a for each a[epsilon]R - {0} which contain no section of type T2, S2, or W[plus sign in circle](W [logical or] W) are classified up to isomorphism by finite vector space subgroups.