Motivic multiple zeta values relative to $\mu_2$

We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to [math] . We find a basis for the depth-graded motivic double zeta values relative to [math] of even weight and a basis for the depth-graded motivic triple zeta values relative to [math] of odd weight. As an application of our main results, we prove Kaneko and Tasaka’s conjectures about the sum odd double zeta values and the classical double zeta values. We also prove an analogue of Kaneko and Tasaka’s conjecture in depth three. Finally, we formulate a conjecture which is related to sum odd multiple zeta values in higher depth.