We present an approach to separating the angular momentum (AM) flux of monochromatic light into its spin and orbital parts based on a symmetrized AM tensor <(M) over bar >. When considering the AM flux for a light beam through its cross section and that for an outgoing wave through a spherical surface in the far-field zone, the separation gives the desired results: the spin/orbital AM flux equals the integral of spin/orbital AM density times some weighting factor accounting for energy flux. When applied to Bessel beams, the obtained spin and orbital AM fluxes are exactly the same as those given by the paper (2014 New J. Phys. 16 093037) based on the canonical AM tensor separation. Furthermore, from the spin AM flux integral, the divergence-free spin AM tensor <(S) over bar > can be identified. We define the orbital AM tensor to be the difference between the total AM tensor <(M) over bar > and <(S) over bar >. Since <(M) over bar > is divergence-free, the integral for either spin and orbital AM flux, can be made on any closed surface.