  We prove a structure theorem for finite-dimensional indefinite-signature
metric 3-Lie algebras admitting a maximally isotropic centre. This algebraic
condition indicates that all the negative-norm states in the associated
Bagger-Lambert theory can be consistently decoupled from the physical Hilbert
space. As an immediate application of the theorem, new examples beyond index 2
are constructed. The lagrangian for the Bagger-Lambert theory based on a
general physically admissible 3-Lie algebra of this kind is obtained. Following
an expansion around a suitable vacuum, the precise relationship between such
theories and certain more conventional maximally supersymmetric gauge theories
is found. These typically involve particular combinations of N=8 super
Yang-Mills and massive vector supermultiplets. A dictionary between the 3-Lie
algebraic data and the physical parameters in the resulting gauge theories will
thereby be provided.
