We discuss two conditions needed for the correct computation of the 2νββ nuclear matrix elements within the realistic shell-model framework. An algorithm in which intermediate states are treated based on Whitehead's moment method is inspected by taking examples of the double GT + transitions 3 6 Ar → 3 6 S, 5 4 Fe → 5 4 Cr and 5 8 Ni → 5 8 Fe. This algorithm yields rapid convergence on the 2νββ matrix elements, even when neither the relevant GT + nor GT - strength distribution is convergent. A significant role of the shell structure is pointed out, which makes the 2νββ matrix elements highly dominated by the low-lying intermediate states. Experimental information of the low-lying GT ± strengths is strongly desired. Half-lives of T 2 ν 1 2 (EC/EC; 3 6 Ar → 3 6 S) = 1.7 10 2 9 yr, T 2 ν 1 2 (EC/EC; 5 4 Fe → 5 4 Cr) = 1.5 10 2 7 yr, T 1 2 (EC/EC; 5 8 Ni → 5 8 Fe) = 6.1 10 2 4 yr and T 2 ν 1 2 (β + /EC; 5 8 Ni → 5 8 Fe) = 8.6 10 2 5 yr are obtained from the present realistic shell-model calculation of the nuclear matrix elements.