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Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ?

For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie algebras), with :

$$ [T_a, T_b, T_c] = f_{abc}^d T_d $$

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What you describe is not a Lie 2-algebra, it is a Lie triple system (en.wikipedia.org/wiki/Triple_system#Lie_triple_systems). A Baez-Crans Lie 2-algebra is a Lie algebra internal to the category of 2-term chain complexes. –  Qiaochu Yuan Oct 27 '12 at 18:10
@Qiaochu Yuan: Thanks. Is there any relation between the two (Lie triple system and Lie 2-algebra) ? –  Trimok Oct 27 '12 at 18:13
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