We recast physical properties of the Bagger-Lambert theory, such as
shift-symmetry and decoupling of ghosts, the absence of scale and parity
invariance, in Lie 3-algebraic terms, thus motivating the study of metric Lie
3-algebras and their Lie algebras of derivations. We prove a structure theorem
for metric Lie 3-algebras in arbitrary signature showing that they can be
constructed out of the simple and one-dimensional Lie 3-algebras iterating two
constructions: orthogonal direct sum and a new construction called a double
extension, by analogy with the similar construction for Lie algebras. We
classify metric Lie 3-algebras of signature (2,p) and study their Lie algebras
of derivations, including those which preserve the conformal class of the inner
product. We revisit the 3-algebraic criteria spelt out at the start of the
paper and select those algebras with signature (2,p) which satisfy them, as
well as indicate the construction of more general metric Lie 3-algebras
satisfying the ghost-decoupling criterion.Comment: 38 page
