Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations: $$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$ $$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}$$ It appears that these two equations are not equivalent, although I don't know a single example for an $R$ that satisfies one of the equations and not the other.

**Are there known additional conditions on the Hopf algebra that imply the equivalence of the two braid relations?**

Or otherwise:

**Is there a Hopf algebra $H$ and an element $R \in H \otimes H$ that satisfies e.g. the first braid relation all further axioms, but not the second braid relation?**

Motivation: I think I have discovered a case where they are equivalent, which is the case of a *quantum group with half-twist*, as described in this article by Noah Snyder and Peter Tingley. It deals with Hopf algebras where $R = \left(t^{-1} \otimes t^{-1}\right) \Delta(t)$ for a suitable element $t$ in the Hopf algebra which is called the half-twist. I'm currently writing this particular instance up, but I'm very curious whether there are other conditions that make the braid relations equivalent, or counterexamples. It's nice if they are equivalent, since it saves you a lot of work when searching for new examples.