We give an invariant classification of orbits of the fundamental
representations of exceptional groups E7(7) and E6(6) which classify
BPS states in string and M theories toroidally compactified to d=4 and d=5. The
exceptional Jordan algebra and the exceptional Freudenthal triple system and
their cubic and quartic invariants play a major role in this classification.
The cubic and quartic invariants correspond to the black hole entropy in d=5
and d=4, respectively. The classification of BPS states preserving different
numbers of supersymmetries is in close parallel to the classification of the
little groups and the orbits of timelike, lightlike and space-like vectors in
Minkowski space. The orbits of BPS black holes in N=2 Maxwell-Einstein
supergravity theories in d=4 and d=5 with symmetric space geometries are also
classified including the exceptional N=2 theory that has E7(−25) and
E6(−26) as its symmety in the respective dimensions.Comment: New references and two tables added, a new section on the orbits of
N=2 Maxwell-Einstein supergravity theories in d=4 and d=5 included and some
minor changes were made in other sections. 17 pages. Latex fil