We have investigated the intrasubband spin-density excitation (SDE) in an asymmetrically doped GaAs-AlGaAs single quantum well with balanced Rashba and Dresselhaus spin-orbit interaction strengths by inelastic light scattering. For this unique symmetry, the combined spin-orbit field is either parallel or antiparallel to the [1 (1) over bar0] in-plane direction of the quantum well for all wave vectors of the two-dimensional reciprocal space. In backscattering geometry, the SDE is formed by spin-flip intrasubband transitions of the spin-split subband. Via the splitting of the intrasubband SDE, we have directly detected the spin splitting of the conduction band due to the anisotropic spin-orbit field. As expected, the splitting is nonzero if a wave vector is transferred perpendicular to the direction of the spin-orbit field and close to zero for a parallel wave-vector transfer. The extracted values for the spin-orbit strength and for the wavelength of a persistent spin helix compare well with results of previous experiments of direct spatial mapping of the spin helix.