In the meson-theoretical potential model for the study of the nonmesonic decay rates and asymmetries of hypernuclei, for the first time, the axial-vector a(1) meson (J(PC)=1(++),m(a1)=1230 MeV) is introduced. The a(1) meson is the chiral partner of the rho meson and has been treated in the meson-pair exchange framework as rho pi/a(1) and sigma pi/a(1). This is analogous to the treatment of rho and sigma exchange in our model. The a(1)-meson exchange is found to give remarkable modifications of the parity-conserving decay potentials ((1,3)S ->(1,3)S and (3)S(1)->(3)D(1)) at short range r <= 1 fm. As a result, the calculated intrinsic asymmetry parameter alpha(Lambda) for (5)(Lambda)He becomes very small and positive in good agreement with the recent high-quality experimental data. The calculated small values of alpha(Lambda) are well compared with the data for (11)(Lambda)B and (12)(Lambda)C within error bars. The inclusion of the a(1) meson also improves the Gamma(n)/Gamma(p) ratios and leads to a consistent explanation for the existing nonmesonic weak decay data of the light Lambda hypernuclei (A <= 12). The results calculated in the pi+2 pi/rho+2 pi/sigma+omega+K+rho pi/a(1)+sigma pi/a(1) exchange interaction model are presented together with the estimates without a(1). Also, the derivation of the expression for the proton asymmetry is described in some detail to elucidate the calculation procedures and phase conventions.