Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs

Let (U, U⁾ be a split affine quantum symmetric pair of type Bₙ⁽¹⁾, Cₙ⁽¹⁾ or Dₙ⁽¹⁾. We prove factorization and coproduct formulae for the Drinfeld-Cartan operators Θᵢ(z) in the Lu-Wang Drinfeld-type presentation, generalizing the type Aₙ⁽¹⁾ result from [Prz23]. As an application, we show that a boundary analogue of the q-character map, defined via the spectra of these operators, is compatible with the usual q-character map. As an auxiliary result, we also produce explicit reduced expressions for the fundamental weights in the extended affine Weyl groups of classical types.