It is a long-standing conjecture that the Grothendieck-Teichmüller Lie algebra grt is isomorphic to the double shuffle Lie algebra, which is itself conjectured to be dual to the Q-vector space of multiple zeta values. The comparison between these two Lie algebras is difficult due to the difference in their definitions. In this talk I will give a new equivalent definition of grt which is much closer in nature to the description of couble shuffle, and use it to recover Furusho's important theorem proving that grt is included in the double shuffle Lie algebra. The formulas in the new definition of grt are interesting and in particular heavily rely on an algebraic form of renormalisation.