  We give necessary and sufficient conditions for a weight system on multiloop
chord diagrams to be obtainable from a metrized Lie algebra representation, in
terms of a bound on the ranks of associated connection matrices.
  Here a multiloop chord diagram is a graph with directed and undirected edges
so that at each vertex precisely one directed edge is entering and precisely
one directed edge is leaving, and each vertex is incident with precisely one
undirected edge. Weight systems on multiloop chord diagrams yield the Vassiliev
invariants for knots and links.
  The $k$-th connection matrix of a function $f$ on the collection of multiloop
chord diagrams is the matrix with rows and columns indexed by $k$-labeled chord
tangles, and with entries equal to the $f$-value on the join of the tangles.
