  The leading excitations of the dilute $A_L$ model in regime 2 are considered
using analytic arguments. The model can be identified with the integrable
$\phi_{1,2}$ perturbation of the unitary minimal series $M_{L,L+1}$. It is
demonstrated that the excitation spectrum of the transfer matrix satisfies the
same functional equations in terms of elliptic functions as the exact
S-matrices of the $\phi_{1,2}$ perturbation do in terms of trigonometric
functions. In particular, the bootstrap equation corresponding to a self-fusing
process is recovered. For the special cases $L=3,4,6$ corresponding to the
Ising model in a magnetic field, and the leading thermal perturbations of the
tricritical Ising and three-state Potts model, as well as for the unrestricted
model, $L=\infty$, we relate the structure of the Bethe roots to the Lie
algebras $E_{8,7,6}$ and $D_4$ using Coxeter geometry. In these cases Coxeter
geometry also allows for a single formula in generic Lie algebraic terms
describing all four cases. For general $L$ we calculate the spectral gaps
associated with the leading excitation which allows us to compute universal
amplitude ratios characteristic of the universality class. The ratios are of
field theoretic importance as they enter the bulk vacuum expectation value of
the energy momentum tensor associated with the corresponding integrable quantum
field theories.
