Verify the second Serre relation for the sl(2) case

Verify the second Serre relation for the generalized BIZZ charges and establish the corresponding classical Yangian relations for the special case in which the Lie algebra is \(\mathfrak{sl}(2)\), including the real forms \(\mathfrak{su}(2)\) and \(\mathfrak{sl}(2,\mathbb R)\).

Background

The paper applies a generalized BIZZ theorem to the auxiliary-field-deformed lambda model and derives the classical Yangian relations for the first two nonlocal charges. The argument is stated for finite-dimensional simple Lie algebras whose complexification has no $\mathfrak{sl}(2,\mathbb C)$ factor.

The $\mathfrak{sl}(2)$ case is exceptional because the first Serre relation is identically satisfied. Consequently, the proof given in the paper does not establish the remaining, second Serre relation for $\mathfrak{sl}(2)$, leaving verification of the full classical Yangian structure in this case unresolved. The authors explicitly defer this verification to future work.