  We continue the study of stabilization phenomena for Dynkin diagram sequences
initiated in the earlier work of Kleber and the present author. We consider a
more general class of sequences than that of this earlier work, and isolate a
condition on the weights that gives stabilization of tensor product and
branching multiplicities. We show that all the results of the previous article
can be naturally generalized to this setting. We also prove some properties of
the partially ordered set of dominant weights of indefinite Kac-Moody algebras,
and use this to give a more concrete definition of a stable representation
ring. Finally, we consider the classical sequences $B_n, C_n, D_n$ that fall
outside the purview of the earlier work, and work out some simple conditions on
the weights which imply stabilization.
