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As Lie algebras of compact connected Lie groups, semisimple Lie algebras havewide applications in the description of continuous symmetries of physicalsystems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis ofgenerators which consists of a Cartan subalgebra of mutually commutinggenerators H_I and a number of step generators E^\alpha that are characterizedby a root space of non-degenerate one-forms \alpha. This simple decompositionin terms of the root space allows for a complete classification of semisimpleLie algebras. In this paper, we introduce the analogous concept of aCartan-Weyl Lie 3-algebra. We analyze their structure and obtain a completeclassification of them. Many known examples of metric Lie 3-algebras (e.g. theLorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due totheir elegant and simple structure, we speculate that Cartan-Weyl 3-algebrasmay be useful for describing some kinds of generalized symmetries. As anapplication, we consider their use in the Bagger-Lambert-Gustavsson (BLG)theory.

Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras