  In transition metal dichalcogenide semiconductor monolayers the spin dynamics
of electrons is controlled by the original spin-valley locking effect resulting
from the interplay between spin-orbit interaction and inversion asymmetry. As a
consequence, for electrons occupying bottom conduction bands, a carrier spin
flip occurs only if there is a simultaneous change of valley. However, very
little is known about the intra-valley spin relaxation processes. In this work
we have performed stationary and time-resolved photoluminescence measurements
in high quality WSe$_2$ monolayers. Our experiments highlight an efficient
relaxation from bright to dark excitons, due to a fast intra-valley electron
transfer from the top to the bottom conduction band with opposite spins. A
combination of experiments and theoretical analysis allows us to infer a spin
relaxation time of about $\tau_s\sim10~$ps, driven by the interplay between
$\Gamma$-valley chiral phonons and spin-orbit mixing.
