lie algebras - Is this an identity in Lie bialgebras?

Perhaps this will be a trivial question. For this post, everything is over your favorite field of characteristic $0$.

Definitions and notation

Recall that a Lie algebra is a vector space $mathfrak g$ along with a map $beta: mathfrak g^{wedge 2} to mathfrak g$ satisfying the Jacobi identity. One way to write the Jacobi identity is as follows: extend $beta$ to $mathfrak g^{otimes 2} to mathfrak g$ via the usual projection $mathfrak g^{otimes 2} to mathfrak g^{wedge 2}$, consider the map $beta circ (1 otimes beta): mathfrak g^{otimes 3} to mathfrak g$; then the restriction of this map to $mathfrak g^{wedge 3} subseteq mathfrak g^{otimes 3}$ vanishes. (Because $beta$ vanishes on the symmetric product $mathfrak g^{vee 2}$, the Jacobi identity is equivalent to $beta circ (1 otimes beta)$ vanishing on $mathfrak g^{vee 3}$.)

A Lie coalgebra is a vector space $mathfrak g$ with a map $delta: mathfrak g to mathfrak g^{wedge 2}$, satisfying the coJacobi identity, which asserts that the map $(delta otimes 1) circ delta: mathfrak g to mathfrak g^{wedge 3}$ vanishes. A vector space $mathfrak g$ that is both a Lie algebra (under $beta$) and a Lie coalgebra (under $delta$), is a Lie bialgebra if $beta$ and $delta$ satisfy an additional relationship. Namely, let $sigma: mathfrak g^{otimes 2} to mathfrak g^{otimes 2}$ be the usual "flip" map; then the bialgebra identity is that $delta circ beta$ and $(1 otimes beta)circ (delta otimes 1) + (beta otimes 1) circ (1 otimes delta) + (beta otimes 1) circ (1otimes sigma) circ (delta otimes 1) + (1 otimes beta) circ (sigma otimes 1) circ (1 otimes delta)$ are equal as maps $mathfrak g^{otimes 2} to mathfrak g^{otimes 2}$.

My question

In a calculation I'm doing, I'm led to consider the map $mathfrak g^{otimes 2} to mathfrak g^{vee 3}$ given by $(1 otimes beta otimes 1) circ (delta otimes delta)$. (I mean, $(1 otimes beta otimes 1) circ (delta otimes delta)$ lands in $mathfrak g^{otimes 3}$, but I want the composition with the natural projection $mathfrak g^{otimes 3} to mathfrak g^{vee 3}$.) In particular, for the calculation to come out right, I'd like for this map to vanish. Does it?

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