  For odd $N\geq 5$, we establish a short exact sequence about motivic double
zeta values $\zeta^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. From
this we classify all the relations among depth-graded motivic double zeta
values $\zeta^{\mathfrak{m}}(r,N-r)$ with $r\geq3$ odd, $N-r\geq2$. 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.
