From the introduction: “In [Automorphic forms and zeta functions. In memory of Tsuneo Arakawa. Proc. Conf., Rikkyo University, Tokyo, Japan, 2004. Hackensack, NJ: World Scientific, 71–106 (2006; Zbl 1122.11057)], H. Gangl, D. Zagier and the first author studied in detail the “double shuffle relations” satisfied by the double zeta values \[ \zeta(r, s) =\sum_{m>n>0} \frac 1{m^r n^s}\quad (r\geq 2, s\geq 1), \tag{1} \] and revealed, in particular, various connections between the space of double zeta values and the space of modular forms as well as their period polynomials on the full modular group \(\mathrm{PSL}_2(\mathbb Z)\). They also defined the “double Eisenstein series” and deduced the double shuffle relations for them. In [Proc. Japan-Korea joint seminar on number theory, 2004, Kuju, Japan. Fukuoka: Kyushu University, 79–85 (2004; Zbl 1071.11052)] the first author illustrated a way to connect the double Eisenstein series to the period polynomials of modular forms (of level 1).”
“In the present paper, the authors consider the double shuffle relations of level 2 and study the formal double zeta space, whose generators are the formal symbols corresponding to the double zeta values of level 2 (Euler sums) and the defining relations are the double shuffle relations. One of the relations we obtain in the formal double zeta space (Theorem 1) has an interesting application to the problem of representations of integers as sums of squares, and this will be given in the subsequent paper by the second author [Ramanujan J. 33, No. 1, 1–21 (2014; Zbl 1392.11021)]. ”
“The authors then proceed to define the double Eisenstein series of level 2 and show that they also satisfy the double shuffle relations (Theorem 3), and have connections like in the case of level 1 to double zeta values, modular forms, and period polynomials, of level 2 (Theorem 5 and Corollary 1).” (The theorems are too lengthy to restate here.)