Generators for vector spaces spanned by double zeta values with even weight

Let DZk be the Q-vector space spanned by double zeta values with weight k, and DMk be its quotient space divided by the space PZk spanned by the zeta value ζ(k) and products of two zeta values with total weight k. When k is even, an upper bound for the dimension of DMk is known. By adding the dimensions of DMk and PZk, an upper bound of DZk which equals k/2 minus the dimension of the space of modular forms of weight k on the modular group is given. In this note, we obtain some specific sets of generators for DMk which represent the upper bound. These yield the corresponding sets and the upper bound for DZk.