  We present fermionic quasi-particle sum representations for some of the
characters (or branching functions) of ~${(G^{(1)})_1 \times (G^{(1)})_1 \o
(G^{(1)})_2}$ ~for all simply-laced Lie algebras $G$. For given $G$ the
characters are written as the partition function of a set of rank~$G$ types of
massless quasi-particles in certain charge sectors, with nontrivial lower
bounds on the one-particle momenta. We discuss the non-uniqueness of the
representations for the identity character of the critical Ising model, which
arises in both the $A_1$ and $E_8$ cases.
