Motivic double zeta values of odd weight

For odd $$N\ge 5$$ N ≥ 5 , we establish a short exact sequence about motivic double zeta values $$\zeta ^{\mathfrak {m}}(r,N-r)$$ ζ m ( r , N - r ) with $$r\ge 3$$ r ≥ 3 odd, $$N-r\ge 2$$ N - r ≥ 2 . From this we classify all the relations among depth-graded motivic double zeta values $$\zeta ^{\mathfrak {m}}(r,N-r)$$ ζ m ( r , N - r ) with $$r\ge 3$$ r ≥ 3 odd, $$N-r\ge 2$$ N - r ≥ 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.