  We study the eigenspace decomposition of a basic classical Lie superalgebra
under the adjoint action of a toral subalgebra, thus extending results of
Kostant. In recognition of Kostant's contribution we refer to the eigenspaces
appearing in the decomposition as Kostant roots. We then prove that Kostant
root systems inherit the main properties of classical root systems. Our
approach is combinatorial in nature and utilizes certain graphs naturally
associated with Kostant root systems. In particular, we reprove Kostant's
results without making use of the Killing form.
