Motivic double zeta values of odd weight

For odd $N\ge 5$, we establish a short exact sequence about motivic double zeta values ${\zeta }^{\mathfrak{m}}(r,N-r)$ with $r\ge 3$ odd, $N-r\ge 2$. From this we classify all the relations among depth-graded motivic double zeta values ${\zeta }^{\mathfrak{m}}(r,N-r)$ with $r\ge 3$ odd, $N-r\ge 2$. As a corollary, we confirm a conjecture of Zagier on the rank of a matrix which concerns relations among multiple zeta values of odd weight.