  We give an invariant classification of orbits of the fundamental
representations of exceptional groups $E_{7(7)}$ and $E_{6(6)}$ 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 $E_{7(-25)}$ and
$E_{6(-26)}$ as its symmety in the respective dimensions.
