  We classify canonical metric 3-algebra structures on matrix algebras and find
novel three-dimensional conformally invariant actions in N=4 projective
superspace based on them. These can be viewed as Chern-Simons theories with
special matter content and couplings. We explore the relations of these actions
with the N=2 actions based on generalized 3-Lie algebras found earlier and
relate them to the original Bagger-Lambert-Gustavsson action.
