  The purpose of the present paper is twofold. In the first part, we provide an
algebraic characterization of several families of $\nu= \frac{1}{2^n}$ $n\leq
5$ BPS states in M theory, at threshold and non-threshold, by an analysis of
the BPS bound derived from the ${\cal N}=1$ D=11 SuperPoincar\'e algebra. We
determine their BPS masses and their supersymmetry projection conditions,
explicitly. In the second part, we develop an algebraic formulation to study
the way BPS states transform under $GL(32,\bR)$ transformations, the group of
automorphisms of the corresponding SuperPoincar\'e algebra. We prove that all
$\nu={1/2}$ non-threshold bound states are SO(32) related with $\nu={1/2}$ BPS
states at threshold having the same mass. We provide further examples of this
phenomena for less supersymmetric $\nu={1/4},{1/8}$ non-threshold bound states.
