  One of the most important questions in the Bagger-Lambert-Gustavsson (BLG)
theory of multiple M2-branes is the choice of the Lie 3-algebra. The Lie
3-algebra should be chosen such that the corresponding BLG model is unitary and
admits fuzzy 3-sphere as a solution. In this paper we propose another new
condition: the Lie 3-algebras of use must be connected to the semisimple Lie
algebras describing the gauge symmetry of D-branes via a certain reduction
condition. We show that this reduction condition leads to a natural
generalization of the Cartan-Weyl 3-algebras introduced in arXiv:1004.1397.
Similar to a Cartan-Weyl 3-algebra, a generalized Cartan-Weyl 3-algebra
processes a set of step generators characterized by non-degenerate roots.
However, its Cartan subalgebra is non-abelian in general. We give reasons why
having a non-abelian Cartan subalgebra may be just right to allow for fuzzy
3-sphere solution in the corresponding BLG models. We propose that generalized
Cartan-Weyl 3-algebras is the right class of metric Lie 3-algebras to be used
in the BLG theory.
