  We report Electron Spin Resonance (ESR) measurements on stage-I potassium
intercalated graphite (KC$_8$). Angular dependent measurements show that the
spin-lattice relaxation time is longer when the magnetic field is perpendicular
to the graphene layer as compared to when the magnetic field is in the plane.
This anisotropy is analyzed in the framework of the Elliott-Yafet theory of
spin-relaxation in metals. The analysis considers an anisotropic spin-orbit
Hamiltonian and the first order perturbative treatment of Elliott is reproduced
for this model Hamiltonian. The result provides an experimental input for the
first-principles theories of spin-orbit interaction in layered carbon and thus
to a better understanding of spin-relaxation phenomena in graphene and in other
layered materials as well.
