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We prove a structure theorem for finite-dimensional indefinite-signaturemetric 3-Lie algebras admitting a maximally isotropic centre. This algebraiccondition indicates that all the negative-norm states in the associatedBagger-Lambert theory can be consistently decoupled from the physical Hilbertspace. As an immediate application of the theorem, new examples beyond index 2are constructed. The lagrangian for the Bagger-Lambert theory based on ageneral physically admissible 3-Lie algebra of this kind is obtained. Followingan expansion around a suitable vacuum, the precise relationship between suchtheories and certain more conventional maximally supersymmetric gauge theoriesis found. These typically involve particular combinations of N=8 superYang-Mills and massive vector supermultiplets. A dictionary between the 3-Liealgebraic data and the physical parameters in the resulting gauge theories willthereby be provided.

Metric 3-Lie algebras for unitary Bagger-Lambert theories